Question: What are the input variables of a function sometimes referred to as?

Answer: The output of a function f corresponding to an input x is denoted by

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. <hl> The output of a function f corresponding to an input x is denoted by <hl> 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 would one denote the output of a function 'f' corresponding to an input 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. 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: f(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 <hl> f(x) <hl> (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: What are considered the "central objects of investigation" in most fields of mathematics?

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Answer: Functions

<hl> Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. <hl> There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

<hl> Functions <hl> of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.


Question: If a function is presented as a picture, what would it be called?

Answer: the graph of the function

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. <hl> Others are given by a picture, called the graph of the function . <hl> In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called <hl> the graph of the function <hl>. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Others are given by a picture, called <hl> the graph of the function <hl> .


Question: When functions are defined in science by a list of outputs and their inputs. what would this be called?

Answer: a table

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. <hl> In science, functions are sometimes defined by a table that gives the outputs for selected inputs. <hl> A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by <hl> a table <hl> that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.


Question: What is the name for a system that tells how to compute the output for a given input?

Answer: a formula or algorithm

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. <hl> Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. <hl> Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.

Functions of various kinds are "the central objects of investigation" in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by <hl> a formula or algorithm <hl> that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation.


Question: What is the name for a function's set of inputs?

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2).  The image of this function is the set of non-negative real numbers.  Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

Answer: domain

<hl> In modern mathematics, a function is defined by its set of inputs, called the domain ; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. <hl> Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the <hl> domain <hl>; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the <hl> domain <hl> ; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph.


Question: What is the name for a function's set of outputs?

Answer: codomain

<hl> In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain ; and the set of all input-output pairs, called its graph. <hl> Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its <hl> codomain <hl>; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its <hl> codomain <hl> ; and the set of all input-output pairs, called its graph.


Question: What is the name of all input-output pairs of a function?

Answer: graph

<hl> In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph . <hl> Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its <hl> graph <hl>. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its <hl> graph <hl> .


Question: What is commonly used to refer to a function's outputs, but is also sometimes used to refer to the function's codomain?

Answer: range

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. <hl> Sometimes the codomain is called the function's " range ", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). <hl> For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "<hl> range <hl>", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

Sometimes the codomain is called the function's " <hl> range <hl> ", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function).


Question: What is the name for collections of functions with the same domain and the same codomain?

Answer: function spaces

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. <hl> Collections of functions with the same domain and the same codomain are called function spaces , the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis. <hl>

In modern mathematics, a function is defined by its set of inputs, called the domain; a set containing the set of outputs, and possibly additional elements, as members, called its codomain; and the set of all input-output pairs, called its graph. Sometimes the codomain is called the function's "range", but more commonly the word "range" is used to mean, instead, specifically the set of outputs (this is also called the image of the function). For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the graph consists of all pairs of real numbers (x, x2). The image of this function is the set of non-negative real numbers. Collections of functions with the same domain and the same codomain are called <hl> function spaces <hl>, the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.

Collections of functions with the same domain and the same codomain are called <hl> function spaces <hl> , the properties of which are studied in such mathematical disciplines as real analysis, complex analysis, and functional analysis.


Question: For the example given, what four shapes does X consist of?

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color.  There is no shape that lacks a color and no shape that has two or more colors.  This function will be referred to as the "color-of-the-shape function".

Answer: a red triangle, a yellow rectangle, a green hexagon, and a red square

<hl> For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square ; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. <hl> Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function".

For an example of a function, let X be the set consisting of four shapes: <hl> a red triangle, a yellow rectangle, a green hexagon, and a red square <hl>; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function".

For an example of a function, let X be the set consisting of four shapes: <hl> a red triangle, a yellow rectangle, a green hexagon, and a red square <hl> ; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow.


Question: For the example given, what five colors does Y consist of?

Answer: "color-of-the-shape function"

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. <hl> This function will be referred to as the "color-of-the-shape function" . <hl>

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the <hl> "color-of-the-shape function" <hl>.

This function will be referred to as the <hl> "color-of-the-shape function" <hl> .


Question: When linking the shapes and colors of X and Y, what does the example refer to this function as?

Answer: the "color-of-the-shape function"

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. <hl> This function will be referred to as the "color-of-the-shape function" . <hl>

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as <hl> the "color-of-the-shape function" <hl>.

This function will be referred to as <hl> the "color-of-the-shape function" <hl> .


Question: In the example given, how many colors was each shape linked to?

Answer: to exactly one color

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. <hl> Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", to exactly one color . <hl> There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function".

For an example of a function, let X be the set consisting of four shapes: a red triangle, a yellow rectangle, a green hexagon, and a red square; and let Y be the set consisting of five colors: red, blue, green, pink, and yellow. Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", <hl> to exactly one color <hl>. There is no shape that lacks a color and no shape that has two or more colors. This function will be referred to as the "color-of-the-shape function".

Linking each shape to its color is a function from X to Y: each shape is linked to a color (i.e., an element in Y), and each shape is "linked", or "mapped", <hl> to exactly one color <hl> .


Question: What is the input to a function called?

The input to a function is called the argument and the output is called the value. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

Answer: the argument

<hl> The input to a function is called the argument and the output is called the value. <hl> The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The input to a function is called <hl> the argument <hl> and the output is called the value. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.


Question: What is the name for the set of all permissible outputs for a given function?

Answer: codomain

The input to a function is called the argument and the output is called the value. <hl> The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain . <hl> Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The input to a function is called the argument and the output is called the value. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the <hl> codomain <hl>. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the <hl> codomain <hl> .


Question: What is the name for the set of all permitted inputs for a given function?

Answer: domain

The input to a function is called the argument and the output is called the value. <hl> The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. <hl> Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The input to a function is called the argument and the output is called the value. The set of all permitted inputs to a given function is called the <hl> domain <hl> of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.


Question: What is the output to a function called?

Answer: value

<hl> The input to a function is called the argument and the output is called the value . <hl> The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The input to a function is called the argument and the output is called the <hl> value <hl>. The set of all permitted inputs to a given function is called the domain of the function, while the set of permissible outputs is called the codomain. Thus, the domain of the "color-of-the-shape function" is the set of the four shapes, and the codomain consists of the five colors. The concept of a function does not require that every possible output is the value of some argument, e.g. the color blue is not the color of any of the four shapes in X.

The input to a function is called the argument and the output is called the <hl> value <hl> .


Question: How many different ways can "a function from X to Y" be presented?

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.

Answer: two different ways

<hl> The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. <hl> Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are <hl> two different ways <hl> a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.


Question: What is X called in the function?

Answer: domain

<hl> The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. <hl> Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the <hl> domain <hl> X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.


Question: What is Y called in the function?

Answer: codomain

<hl> The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. <hl> Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and <hl> codomain <hl> Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the codomain is not specified, and in the other it is a valid definition.


Question: What makes f(x) = x2 different?

Answer: codomain is not specified

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 <hl> on the reals does not completely specify a function as the codomain is not specified , and in the other it is a valid definition. <hl>

The above definition of "a function from X to Y" is generally agreed on,[citation needed] however there are two different ways a "function" is normally defined where the domain X and codomain Y are not explicitly or implicitly specified. Usually this is not a problem as the domain and codomain normally will be known. With one definition saying the function defined by f(x) = x2 on the reals does not completely specify a function as the <hl> codomain is not specified <hl>, and in the other it is a valid definition.

on the reals does not completely specify a function as the <hl> codomain is not specified <hl> , and in the other it is a valid definition.


Question: Define one type of function?

If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y.  This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.

Answer: set of ordered pairs with no specific codomain

<hl> If a function is defined as a set of ordered pairs with no specific codomain , then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. <hl> Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.

If a function is defined as a <hl> set of ordered pairs with no specific codomain <hl>, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.

If a function is defined as a <hl> set of ordered pairs with no specific codomain <hl> , then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard.


Question: How can f:X -> Y be referred to?

Answer: f maps X into Y

If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. <hl> This is also referred to by saying that " f maps X into Y " In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. <hl> With the definition of a function as an ordered triple this would always be considered a partial function.

If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "<hl> f maps X into Y <hl>" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.

This is also referred to by saying that " <hl> f maps X into Y <hl> " In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2].


Question: Can you write a function that is a function whose domain is X and image is a subset of Y?

Answer: f: X → Y

<hl> If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. <hl> Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.

If a function is defined as a set of ordered pairs with no specific codomain, then <hl> f: X → Y <hl> indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an ordered triple this would always be considered a partial function.


Question: What type of function would f:X -> Y be considered a partial function?

Answer: ordered triple

If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. <hl> With the definition of a function as an ordered triple this would always be considered a partial function. <hl>

If a function is defined as a set of ordered pairs with no specific codomain, then f: X → Y indicates that f is a function whose domain is X and whose image is a subset of Y. This is the case in the ISO standard. Y may be referred to as the codomain but then any set including the image of f is a valid codomain of f. This is also referred to by saying that "f maps X into Y" In some usages X and Y may subset the ordered pairs, e.g. the function f on the real numbers such that y=x2 when used as in f: [0,4] → [0,4] means the function defined only on the interval [0,2]. With the definition of a function as an <hl> ordered triple <hl> this would always be considered a partial function.


Question: How are functions commonly defined?

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Answer: a type of relation

<hl> Functions are commonly defined as a type of relation . <hl> A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Functions are commonly defined as <hl> a type of relation <hl>. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Functions are commonly defined as <hl> a type of relation <hl> .


Question: How is a relation defined?

Answer: a set of ordered pairs

Functions are commonly defined as a type of relation. <hl> A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. <hl> However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Functions are commonly defined as a type of relation. A relation from X to Y is <hl> a set of ordered pairs <hl> (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.


Question: How is correspondence defined?

Answer: triple (X, Y, F)

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. <hl> Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F) , however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. <hl> The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a <hl> triple (X, Y, F) <hl>, however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a <hl> triple (X, Y, F) <hl> , however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets.


Question: What is a function as a triple a type of?

Answer: a type of correspondence

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. <hl> The definition of a function as a triple defines a function as a type of correspondence , whereas the definition of a function as a set of ordered pairs defines a function as a type of relation. <hl>

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as <hl> a type of correspondence <hl>, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

The definition of a function as a triple defines a function as <hl> a type of correspondence <hl> , whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.


Question: What is a function as a ordered pair a type of?

Answer: correspondence

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. <hl> Normally a relation is just defined as a set of ordered pairs and a correspondence is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. <hl> The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.

Functions are commonly defined as a type of relation. A relation from X to Y is a set of ordered pairs (x, y) with x ∈ X and y ∈ Y. A function from X to Y can be described as a relation from X to Y that is left-total and right-unique. However, when X and Y are not specified there is a disagreement about the definition of a relation that parallels that for functions. Normally a relation is just defined as a set of ordered pairs and a <hl> correspondence <hl> is defined as a triple (X, Y, F), however the distinction between the two is often blurred or a relation is never referred to without specifying the two sets. The definition of a function as a triple defines a function as a type of correspondence, whereas the definition of a function as a set of ordered pairs defines a function as a type of relation.


Question: What is a type of partial function?

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations.  For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only a partial function from the real line to the real line.  The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

Answer: single-valued relations

<hl> In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations . <hl> For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., <hl> single-valued relations <hl>. For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., <hl> single-valued relations <hl> .


Question: Studying partial functions helps in what types of math?

Answer: recursion theory and functional analysis

<hl> In some parts of mathematics, including recursion theory and functional analysis , it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. <hl> For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including <hl> recursion theory and functional analysis <hl>, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including <hl> recursion theory and functional analysis <hl> , it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations.


Question: What is the value of X in the function f(x) = 1/x?

Answer: x = 0

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. <hl> For example, the function f such that f(x) = 1/x does not define a value for x = 0 , since division by zero is not defined. <hl> Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for <hl> x = 0 <hl>, since division by zero is not defined. Hence f is only a partial function from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

For example, the function f such that f(x) = 1/x does not define a value for <hl> x = 0 <hl> , since division by zero is not defined.


Question: In the function f(x) = 1/x, what is f?

Answer: a partial function

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. <hl> Hence f is only a partial function from the real line to the real line. <hl> The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.

In some parts of mathematics, including recursion theory and functional analysis, it is convenient to study partial functions in which some values of the domain have no association in the graph; i.e., single-valued relations. For example, the function f such that f(x) = 1/x does not define a value for x = 0, since division by zero is not defined. Hence f is only <hl> a partial function <hl> from the real line to the real line. The term total function can be used to stress the fact that every element of the domain does appear as the first element of an ordered pair in the graph.


Question: What is another name for homomorphisms?

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

Answer: structure-preserving functions

<hl> The idea of structure-preserving functions , or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. <hl> In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of <hl> structure-preserving functions <hl>, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of <hl> structure-preserving functions <hl> , or homomorphisms, led to the abstract notion of morphism, the key concept of category theory.


Question: What is morphism a key concept of?

Answer: category theory

<hl> The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory . <hl> In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of <hl> category theory <hl>. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of <hl> category theory <hl> .


Question: What type of function is X x Y?

Answer: well-defined function

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. <hl> However, this is still a well-defined function . <hl> Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a <hl> well-defined function <hl>. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

However, this is still a <hl> well-defined function <hl> .


Question: What function  can f: X -> Y be?

Answer: empty function

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. <hl> Such a function is called an empty function . <hl> In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an <hl> empty function <hl>. In particular, the identity function of the empty set is defined, a requirement for sets to form a category.

Such a function is called an <hl> empty function <hl> .


Question: Why is a identity function of an empty set defined?

Answer: requirement for sets to form a category

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. <hl> In particular, the identity function of the empty set is defined, a requirement for sets to form a category . <hl>

The idea of structure-preserving functions, or homomorphisms, led to the abstract notion of morphism, the key concept of category theory. In fact, functions f: X → Y are the morphisms in the category of sets, including the empty set: if the domain X is the empty set, then the subset of X × Y describing the function is necessarily empty, too. However, this is still a well-defined function. Such a function is called an empty function. In particular, the identity function of the empty set is defined, a <hl> requirement for sets to form a category <hl>.

In particular, the identity function of the empty set is defined, a <hl> requirement for sets to form a category <hl> .


Question: What is (0,0) called on a Cartesian coordinate system?

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Answer: ordered pair

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at <hl> ordered pair <hl> (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


Question: What are the coordinates such as (3,1) called?

Answer: ordered pair

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at <hl> ordered pair <hl> (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


Question: What is the spot an ordered pair points out called?

Answer: at ordered pair (0, 0)

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0) . <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually <hl> at ordered pair (0, 0) <hl>. The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually <hl> at ordered pair (0, 0) <hl> .


Question: What do the numbers in an ordered pair mean?

Answer: signed distances from the origin

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin . <hl>

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as <hl> signed distances from the origin <hl>.

The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as <hl> signed distances from the origin <hl> .


Question: What coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates?

Answer: Cartesian

<hl> A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A <hl> Cartesian <hl> coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


Question: What is the point they meet at called?

Answer: its origin

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is <hl> its origin <hl>, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is <hl> its origin <hl> , usually at ordered pair (0, 0).


Question: How many axes are involved?

Answer: two

<hl> A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from <hl> two <hl> fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


Question: What does the ordered pair represent?

Answer: its origin

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin , usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is <hl> its origin <hl>, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is <hl> its origin <hl> , usually at ordered pair (0, 0).


Question: What are two different names for the type of axis involved?

Answer: coordinate axis or just axis

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. <hl> Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). <hl> The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. Each reference line is called a <hl> coordinate axis or just axis <hl> of the system, and the point where they meet is its origin, usually at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.


Question: Any point in a three dimensional space can be specified by what?

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

Answer: Cartesian coordinates

<hl> One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates , its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). <hl> In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

One can use the same principle to specify the position of any point in three-dimensional space by three <hl> Cartesian coordinates <hl>, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

One can use the same principle to specify the position of any point in three-dimensional space by three <hl> Cartesian coordinates <hl> , its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines).


Question: How many Cartesian coordinates specify a three dimensional space?

Answer: n Cartesian coordinates

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). <hl> In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. <hl>

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, <hl> n Cartesian coordinates <hl> (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.


Question: What type of planes are involved?

Answer: perpendicular

<hl> One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). <hl> In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually <hl> perpendicular <hl> planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.


Question: How many perpendicular planes are involved?

Answer: three mutually perpendicular planes

<hl> One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). <hl> In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.

One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to <hl> three mutually perpendicular planes <hl> (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, n Cartesian coordinates (an element of real n-space) specify the point in an n-dimensional Euclidean space for any dimension n. These coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes.


Question: Who invented Cartesian planes?

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

Answer: René Descartes

<hl> The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. <hl> Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

The invention of Cartesian coordinates in the 17th century by <hl> René Descartes <hl> (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.


Question: When were Cartesian planes invented?

Answer: 17th century

<hl> The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. <hl> Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

The invention of Cartesian coordinates in the <hl> 17th century <hl> by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.


Question: What can Cartesian equations describe?

Answer: geometric shapes (such as curves)

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. <hl> Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. <hl> For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, <hl> geometric shapes (such as curves) <hl> can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.


Question: What type of equations are Cartesian?

Answer: algebraic

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. <hl> Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. <hl> For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.

The invention of Cartesian coordinates in the 17th century by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: <hl> algebraic <hl> equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4.


Question: What is the foundation of analytic geometry?

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

Answer: Cartesian coordinates

<hl> Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. <hl> A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

<hl> Cartesian coordinates <hl> are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.


Question: Cartesian coordinates are the foundation of what type of geometry?

Answer: analytic

<hl> Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. <hl> A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

Cartesian coordinates are the foundation of <hl> analytic <hl> geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.


Question: What is a familiar example of Cartesian coordinates?

Answer: the concept of the graph of a function

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. <hl> A familiar example is the concept of the graph of a function . <hl> Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is <hl> the concept of the graph of a function <hl>. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

A familiar example is <hl> the concept of the graph of a function <hl> .


Question: What is the most common coordinate system in computer graphics?

Answer: Cartesian coordinates

<hl> Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. <hl> A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.

<hl> Cartesian coordinates <hl> are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing.