Question: By what means, does the Classical Association encourage the study of the antiquity?

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

Answer: publications and grants

<hl> In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants . <hl> The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as <hl> publications and grants <hl>. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as <hl> publications and grants <hl> .


Question: What charity in the UK runs a Latin course?

Answer: Via Facilis, a London-based charity

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. <hl> The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity , run Latin courses. <hl> In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and <hl> Via Facilis, a London-based charity <hl>, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and <hl> Via Facilis, a London-based charity <hl> , run Latin courses.


Question: What organization promotes study of the classics in the US and Canada?

Answer: National Junior Classical League

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. <hl> Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. <hl> The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the <hl> National Junior Classical League <hl> (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.


Question: How many members does the national Junior Classical League have?

Answer: more than 50,000 members

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. <hl> Its subsidiaries include the National Junior Classical League (with more than 50,000 members ), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. <hl> The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, the American Classical League supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with <hl> more than 50,000 members <hl>), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

Its subsidiaries include the National Junior Classical League (with <hl> more than 50,000 members <hl> ), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college.


Question: Who sponsors the National Latin Exam?

Answer: the American Classical League

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. <hl> In the United States and Canada, the American Classical League supports every effort to further the study of classics. <hl> Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.

In the United Kingdom, the Classical Association encourages the study of antiquity through various means, such as publications and grants. The University of Cambridge, the Open University (OU), a number of prestigious independent schools, for example Eton and Harrow, and Via Facilis, a London-based charity, run Latin courses. In the United States and Canada, <hl> the American Classical League <hl> supports every effort to further the study of classics. Its subsidiaries include the National Junior Classical League (with more than 50,000 members), which encourages high school students to pursue the study of Latin, and the National Senior Classical League, which encourages students to continue their study of the classics into college. The league also sponsors the National Latin Exam. Classicist Mary Beard wrote in The Times Literary Supplement in 2006 that the reason for learning Latin is because of what was written in it.


Question: According to legends, when was Latin established?

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

Answer: some time before the Trojan War

<hl> According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War .[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. <hl> There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

According to Roman Mythology, Latin was established by a tribal people called the Latini <hl> some time before the Trojan War <hl>.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

According to Roman Mythology, Latin was established by a tribal people called the Latini <hl> some time before the Trojan War <hl> .[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax.


Question: What can be concluded about the rules of classification of languages?

Answer: There are no hard and fast rules of classification

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. <hl> There are no hard and fast rules of classification ; different scholars emphasize different features. <hl> As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. <hl> There are no hard and fast rules of classification <hl>; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

<hl> There are no hard and fast rules of classification <hl> ; different scholars emphasize different features.


Question: Why does the list of historical phrases have variants?

Answer: different scholars emphasize different features

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. <hl> There are no hard and fast rules of classification; different scholars emphasize different features . <hl> As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; <hl> different scholars emphasize different features <hl>. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

There are no hard and fast rules of classification; <hl> different scholars emphasize different features <hl> .


Question: How are the historical phrases distinguished?

Answer: by subtle differences

<hl> According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. <hl> There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished <hl> by subtle differences <hl> in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church, as well as by Protestant scholars, from Late Antiquity onward.


Question: How might the Ecclesiastical Latin be described?

Answer: the styles used by the writers of the Roman Catholic Church

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. <hl> In addition to the historical phases, Ecclesiastical Latin refers to the styles used by the writers of the Roman Catholic Church , as well as by Protestant scholars, from Late Antiquity onward. <hl>

According to Roman Mythology, Latin was established by a tribal people called the Latini some time before the Trojan War.[citation needed] A number of historical phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, morphology, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to <hl> the styles used by the writers of the Roman Catholic Church <hl>, as well as by Protestant scholars, from Late Antiquity onward.

In addition to the historical phases, Ecclesiastical Latin refers to <hl> the styles used by the writers of the Roman Catholic Church <hl> , as well as by Protestant scholars, from Late Antiquity onward.


Question: What is not true of vernacular Latin?

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

Answer: the speech was uniform either diachronically or geographically

<hl> As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically . <hl> On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As vernacular Latin was free to develop on its own, there is no reason to suppose that <hl> the speech was uniform either diachronically or geographically <hl>. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As vernacular Latin was free to develop on its own, there is no reason to suppose that <hl> the speech was uniform either diachronically or geographically <hl> .


Question: What led to the deterioration of the education in Europe in the late Latin era?

Answer: The Decline of the Roman Empire

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. <hl> The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. <hl> This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. <hl> The Decline of the Roman Empire <hl> meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.


Question: Where can we see the post-classical stage of Latin?

Answer: in Christian writings of the time

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. <hl> The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time . <hl> This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen <hl> in Christian writings of the time <hl>. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen <hl> in Christian writings of the time <hl> .


Question: How did Latin change during the post-classical period?

Answer: This language was more in line with the everyday speech

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. <hl> This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses. <hl>

As vernacular Latin was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. <hl> This language was more in line with the everyday speech <hl> not only because of a decline in education, but also because of a desire to spread the word to the masses.


Question: How might have Romanized European populations developed their own dialects of Latin?

Answer: vernacular Latin was free to develop on its own

<hl> As vernacular Latin was free to develop on its own , there is no reason to suppose that the speech was uniform either diachronically or geographically. <hl> On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As <hl> vernacular Latin was free to develop on its own <hl>, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanized European populations developed their own dialects of the language. The Decline of the Roman Empire meant a deterioration in educational standards that brought about Late Latin, a post-classical stage of the language seen in Christian writings of the time. This language was more in line with the everyday speech not only because of a decline in education, but also because of a desire to spread the word to the masses.

As <hl> vernacular Latin was free to develop on its own <hl> , there is no reason to suppose that the speech was uniform either diachronically or geographically.


Question: What theory shows that there are functions that can be precisely defined but are not computable?

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Answer: computability theory

<hl> Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. <hl> Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Fundamental results of <hl> computability theory <hl> show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.


Question: Why is the number of computable functions from integers to integers countable?

Answer: because the number of possible algorithms is

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. <hl> The number of computable functions from integers to integers is countable, because the number of possible algorithms is . <hl> The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, <hl> because the number of possible algorithms is <hl>. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

The number of computable functions from integers to integers is countable, <hl> because the number of possible algorithms is <hl> .


Question: In the sense of cardinality, what are almost all functions from integers to integers?

Answer: not computable

<hl> Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable . <hl> Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Fundamental results of computability theory show that there are functions that can be precisely defined but are <hl> not computable <hl>. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems.

Fundamental results of computability theory show that there are functions that can be precisely defined but are <hl> not computable <hl> .


Question: What is an example of an uncomputable function?

Answer: busy beaver function

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. <hl> Specific examples of uncomputable functions are known, including the busy beaver function and functions related to the halting problem and other undecidable problems. <hl>

Fundamental results of computability theory show that there are functions that can be precisely defined but are not computable. Moreover, in the sense of cardinality, almost all functions from the integers to integers are not computable. The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers. Thus most functions from integers to integers are not computable. Specific examples of uncomputable functions are known, including the <hl> busy beaver function <hl> and functions related to the halting problem and other undecidable problems.


Question: How do you often denote the image of a subset A ⊆ X?

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain.  In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.

Answer: Use of f(A)

<hl> Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. <hl> In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.

<hl> Use of f(A) <hl> to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.


Question: What do some authors use to avoid confusion between the inverse image and inverse function?

Answer: square brackets

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. <hl> Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. <hl> Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use <hl> square brackets <hl> to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.


Question: What would an author write for the preimage of a set and a singleton?

Answer: f−1[B] and f−1[b]

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. <hl> Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton. <hl>

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write <hl> f−1[B] and f−1[b] <hl> for the preimage of a set and a singleton.


Question: When would you be unable to denote the image of a subset A ⊆ X with f(A)?

Answer: the domain is also an element of the domain

<hl> Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of the domain is also an element of the domain . <hl> In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of <hl> the domain is also an element of the domain <hl>. In some fields (e.g., in set theory, where ordinals are also sets of ordinals) it is convenient or even necessary to distinguish the two concepts; the customary notation is f[A] for the set { f(x): x ∈ A }. Likewise, some authors use square brackets to avoid confusion between the inverse image and the inverse function. Thus they would write f−1[B] and f−1[b] for the preimage of a set and a singleton.

Use of f(A) to denote the image of a subset A ⊆ X is consistent so long as no subset of <hl> the domain is also an element of the domain <hl> .


Question: How is the value of x obtained?

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation

g
∘
f

{\displaystyle g\circ f}

, the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order.  The notation can be memorized by reading the notation as "g of f" or "g after f". The composition

g
∘
f

{\displaystyle g\circ f}

 is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order

f
∘
g

{\displaystyle f\circ g}

 need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

Answer: first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y)

<hl> That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y) . <hl> In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. The notation can be memorized by reading the notation as "g of f" or "g after f". The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

That is, the value of x is obtained by <hl> first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y) <hl>. In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. The notation can be memorized by reading the notation as "g of f" or "g after f". The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

That is, the value of x is obtained by <hl> first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y) <hl> .


Question: How can you memorize the notation?

Answer: reading the notation as "g of f" or "g after f"

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. <hl> The notation can be memorized by reading the notation as "g of f" or "g after f" . <hl> The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. The notation can be memorized by <hl> reading the notation as "g of f" or "g after f" <hl>. The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

The notation can be memorized by <hl> reading the notation as "g of f" or "g after f" <hl> .


Question: When is the composition g ∘ f only defined?

Answer: when the codomain of f is the domain of g

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. The notation can be memorized by reading the notation as "g of f" or "g after f". <hl> The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g . <hl> Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order. The notation can be memorized by reading the notation as "g of f" or "g after f". The composition g ∘ f {\displaystyle g\circ f} is only defined <hl> when the codomain of f is the domain of g <hl>. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

The composition g ∘ f {\displaystyle g\circ f} is only defined <hl> when the codomain of f is the domain of g <hl> .


Question: How do you read the notation g ∘ f?

Answer: the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). <hl> In the notation g ∘ f {\displaystyle g\circ f} , the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order . <hl> The notation can be memorized by reading the notation as "g of f" or "g after f". The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

That is, the value of x is obtained by first applying f to x to obtain y = f(x) and then applying g to y to obtain z = g(y). In the notation g ∘ f {\displaystyle g\circ f} , <hl> the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order <hl>. The notation can be memorized by reading the notation as "g of f" or "g after f". The composition g ∘ f {\displaystyle g\circ f} is only defined when the codomain of f is the domain of g. Assuming that, the composition in the opposite order f ∘ g {\displaystyle f\circ g} need not be defined. Even if it is, i.e., if the codomain of f is the codomain of g, it is not in general true that

In the notation g ∘ f {\displaystyle g\circ f} , <hl> the function on the right, f, acts first and the function on the left, g acts second, reversing English reading order <hl> .


Question: What is the overriding of f: X → Y by g: W → Y also called?

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y.  Its graph is the set-theoretical union of the graphs of g and f|X \ W.  Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f.  Overriding is an associative operation; it has the empty function as an identity element.  If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union.  This definition agrees with the definition of union for binary relations.

Answer: overriding union

<hl> The overriding of f: X → Y by g: W → Y (also called overriding union ) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. <hl> If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called <hl> overriding union <hl>) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called <hl> overriding union <hl> ) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element.


Question: How do you denote an extension of g?

Answer: (f ⊕ g): (X ∪ W) → Y

<hl> The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y . <hl> Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as <hl> (f ⊕ g): (X ∪ W) → Y <hl>. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element. If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as <hl> (f ⊕ g): (X ∪ W) → Y <hl> .


Question: What kind of operation is overriding?

Answer: associative operation

<hl> The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation ; it has the empty function as an identity element. <hl> If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an <hl> associative operation <hl>; it has the empty function as an identity element. If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an <hl> associative operation <hl> ; it has the empty function as an identity element.


Question: What does being an associative operation also mean?

Answer: it has the empty function as an identity element

<hl> The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; it has the empty function as an identity element . <hl> If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; <hl> it has the empty function as an identity element <hl>. If f|X ∩ W and g|X ∩ W are pointwise equal (e.g., the domains of f and g are disjoint), then the union of f and g is defined and is equal to their overriding union. This definition agrees with the definition of union for binary relations.

The overriding of f: X → Y by g: W → Y (also called overriding union) is an extension of g denoted as (f ⊕ g): (X ∪ W) → Y. Its graph is the set-theoretical union of the graphs of g and f|X \ W. Thus, it relates any element of the domain of g to its image under g, and any other element of the domain of f to its image under f. Overriding is an associative operation; <hl> it has the empty function as an identity element <hl> .


Question: How do you denote the set of all functions from a set X to a set Y?

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X  → Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

Answer: X → Y, by [X → Y], or by YX

<hl> The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX . <hl> The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

The set of all functions from a set X to a set Y is denoted by <hl> X → Y, by [X → Y], or by YX <hl>. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

The set of all functions from a set X to a set Y is denoted by <hl> X → Y, by [X → Y], or by YX <hl> .


Question: When X and Y are finite and of size |X| and |Y|, then what is the number of functions X  → Y?

Answer: |YX| = |Y||X|

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. <hl> The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X| . <hl> This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is <hl> |YX| = |Y||X| <hl>. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is <hl> |YX| = |Y||X| <hl> .


Question: What field provides notations for sets based on their cardinalities in its convention?

Answer: enumerative combinatorics

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. <hl> This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. <hl> If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. This is an example of the convention from <hl> enumerative combinatorics <hl> that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm.


Question: If X is infinite and there is more than one element in Y, then how many functions are from X to Y?

Answer: uncountably many functions

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. <hl> If X is infinite and there is more than one element in Y then there are uncountably many functions from X to Y, though only countably many of them can be expressed with a formula or algorithm. <hl>

The set of all functions from a set X to a set Y is denoted by X → Y, by [X → Y], or by YX. The latter notation is motivated by the fact that, when X and Y are finite and of size |X| and |Y|, then the number of functions X → Y is |YX| = |Y||X|. This is an example of the convention from enumerative combinatorics that provides notations for sets based on their cardinalities. If X is infinite and there is more than one element in Y then there are <hl> uncountably many functions <hl> from X to Y, though only countably many of them can be expressed with a formula or algorithm.


Question: What is a subset of the Cartesian product X x Y?

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

Answer: function f from X to Y

<hl> A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. <hl> In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

A <hl> function f from X to Y <hl> is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.


Question: How many elements of y are there?

Answer: there is exactly one element y

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. <hl> In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. <hl> This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X <hl> there is exactly one element y <hl> such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.


Question: How many ordered pair in the subset?

Answer: one ordered pair

<hl> A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. <hl> In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only <hl> one ordered pair <hl> in the subset. In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.


Question: What is in X?

Answer: x in X

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. <hl> In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. <hl> This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every <hl> x in X <hl> there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.


Question: What is contained in the subset defining the function of f?

Answer: the ordered pair (x, y) is contained in the subset

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. <hl> In other words, for every x in X there is exactly one element y such that the ordered pair (x, y) is contained in the subset defining the function f. <hl> This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.

A function f from X to Y is a subset of the Cartesian product X × Y subject to the following condition: every element of X is the first component of one and only one ordered pair in the subset. In other words, for every x in X there is exactly one element y such that <hl> the ordered pair (x, y) is contained in the subset <hl> defining the function f. This formal definition is a precise rendition of the idea that to each x is associated an element y of Y, namely the uniquely specified element y with the property just mentioned.


Question: What is a general function often denoted by?

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

Answer: f.

<hl> A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. <hl> Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

A general function is often denoted by <hl> f. <hl> Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.


Question: How is signum function denoted?

Answer: sgn(x)

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. <hl> Given a real number x, its image under the signum function is then written as sgn(x) . <hl> Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as <hl> sgn(x) <hl>. Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

Given a real number x, its image under the signum function is then written as <hl> sgn(x) <hl> .


Question: What is denoted by the symbol x?

Answer: the argument

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). <hl> Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. <hl> For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, <hl> the argument <hl> is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.


Question: What may be omitted when there is little chance of confusion?

Answer: The parentheses around the argument

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). <hl> The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation. <hl>

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted v(t). <hl> The parentheses around the argument <hl> may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.


Question: How is the velocity of some body in physics denoted?

Answer: v(t)

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. <hl> For example, in physics, the velocity of some body, depending on the time, is denoted v(t) . <hl> The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

A general function is often denoted by f. Special functions have names, for example, the signum function is denoted by sgn. Given a real number x, its image under the signum function is then written as sgn(x). Here, the argument is denoted by the symbol x, but different symbols may be used in other contexts. For example, in physics, the velocity of some body, depending on the time, is denoted <hl> v(t) <hl>. The parentheses around the argument may be omitted when there is little chance of confusion, thus: sin x; this is known as prefix notation.

For example, in physics, the velocity of some body, depending on the time, is denoted <hl> v(t) <hl> .


Question: What is often dropped by many authors?

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as

f
(
x
)
=

x

2

−
5
x
+
6

{\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}}

 may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

Answer: the specification of the domain and codomain

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. <hl> Despite that, many authors drop the specification of the domain and codomain , especially if these are clear from the context. <hl> So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop <hl> the specification of the domain and codomain <hl>, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

Despite that, many authors drop <hl> the specification of the domain and codomain <hl> , especially if these are clear from the context.


Question: What kind of context makes it okay to drop the domain and codomain specification?

Answer: clear

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. <hl> Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. <hl> So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are <hl> clear <hl> from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).


Question: What is the formula that agrees?

Answer: f and g

<hl> (with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. <hl> Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

(with different domain) is not considered the same function, even though the formulas defining <hl> f and g <hl> agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).


Question: What would the formula be in the example given?

Answer: f(x) = 4 − x

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. <hl> So in this example many just write f(x) = 4 − x . <hl> Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write <hl> f(x) = 4 − x <hl>. Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

So in this example many just write <hl> f(x) = 4 − x <hl> .


Question: What is also understood implicitly ?

Answer: the maximal possible domain

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. <hl> Sometimes, the maximal possible domain is also understood implicitly: a formula such as f ( x ) <hl> = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).

(with different domain) is not considered the same function, even though the formulas defining f and g agree, and similarly with a different codomain. Despite that, many authors drop the specification of the domain and codomain, especially if these are clear from the context. So in this example many just write f(x) = 4 − x. Sometimes, <hl> the maximal possible domain <hl> is also understood implicitly: a formula such as f ( x ) = x 2 − 5 x + 6 {\displaystyle \scriptstyle f(x)={\sqrt {x^{2}-5x+6}}} may mean that the domain of f is the set of real numbers x where the square root is defined (in this case x ≤ 2 or x ≥ 3).


Question: What is sometimes used as a notation to emphasize the expressions functional nature?

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance,

a
(
⋅

)

2

{\displaystyle \scriptstyle a(\cdot )^{2}}

 stands for the function

x
↦
a

x

2

{\displaystyle \textstyle x\mapsto ax^{2}}

,

∫

a

⋅

f
(
u
)
d
u

{\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du}

 stands for the integral function

x
↦

∫

a

x

f
(
u
)
d
u

{\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du}

, and so on.

Answer: a dot

<hl> To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. <hl> For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes <hl> a dot <hl> notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.


Question: What kind of symbol may be assigned to the variable?

Answer: a special symbol

<hl> To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. <hl> For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning <hl> a special symbol <hl> to the variable. For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.


Question: What stands for the function?

Answer: {\displaystyle \scriptstyle a(\cdot )^{2}}

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. <hl> For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) <hl> d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. For instance, a ( ⋅ ) 2 <hl> {\displaystyle \scriptstyle a(\cdot )^{2}} <hl> stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.


Question: What kind of nature of an expression is talked about?

Answer: functional nature

<hl> To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable. <hl> For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes a dot notation is used in order to emphasize the <hl> functional nature <hl> of an expression without assigning a special symbol to the variable. For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.


Question: A symbol can be used as a what?

Answer: variable

<hl> To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the variable . <hl> For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the <hl> variable <hl>. For instance, a ( ⋅ ) 2 {\displaystyle \scriptstyle a(\cdot )^{2}} stands for the function x ↦ a x 2 {\displaystyle \textstyle x\mapsto ax^{2}} , ∫ a ⋅ f ( u ) d u {\displaystyle \scriptstyle \int _{a}^{\,\cdot }f(u)du} stands for the integral function x ↦ ∫ a x f ( u ) d u {\displaystyle \scriptstyle x\mapsto \int _{a}^{x}f(u)du} , and so on.

To define a function, sometimes a dot notation is used in order to emphasize the functional nature of an expression without assigning a special symbol to the <hl> variable <hl> .


Question: What send integers to integers?

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Answer: Functions

<hl> Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. <hl> Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

<hl> Functions <hl> that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.


Question: What kind of strings can sometimes be defined by an algorithm?

Answer: finite strings to finite strings

<hl> Functions that send integers to integers, or finite strings to finite strings , can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. <hl> Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Functions that send integers to integers, or <hl> finite strings to finite strings <hl>, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Functions that send integers to integers, or <hl> finite strings to finite strings <hl> , can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input.


Question: Functions are definable by an algorithm called what?

Answer: computable functions

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. <hl> Functions definable by an algorithm are called computable functions . <hl> For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called <hl> computable functions <hl>. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.

Functions definable by an algorithm are called <hl> computable functions <hl> .


Question: What kind of algorithm gives a precise process to compute the greatest common divsor of two positive integers?

Answer: Euclidean algorithm

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. <hl> For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers. <hl> Many of the functions studied in the context of number theory are computable.

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the <hl> Euclidean algorithm <hl> gives a precise process to compute the greatest common divisor of two positive integers. Many of the functions studied in the context of number theory are computable.


Question: What kind of integers are used to process and compute the greatest common divisor?

Answer: two positive integers

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. <hl> For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of two positive integers . <hl> Many of the functions studied in the context of number theory are computable.

Functions that send integers to integers, or finite strings to finite strings, can sometimes be defined by an algorithm, which gives a precise description of a set of steps for computing the output of the function from its input. Functions definable by an algorithm are called computable functions. For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of <hl> two positive integers <hl>. Many of the functions studied in the context of number theory are computable.

For example, the Euclidean algorithm gives a precise process to compute the greatest common divisor of <hl> two positive integers <hl> .


Question: What is the name for a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output?

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9.  Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9.  (The same output may be produced by more than one input, but each input gives only one output.)  The input variable(s) are sometimes referred to as the argument(s) of the function.

Answer: function

<hl> In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. <hl> An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

In mathematics, a <hl> function <hl> is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.


Question: How is the term f(x) spoken?

Answer: read "f of x"

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. <hl> The output of a function f corresponding to an input x is denoted by f(x) ( read "f of x" ). <hl> In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (<hl> read "f of x" <hl>). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function.

The output of a function f corresponding to an input x is denoted by f(x) ( <hl> read "f of x" <hl> ).