hcju/proofwikips huggingface dataset:
Definition:Kernel (Abstract Algebra)
=== Kernel of Magma Homomorphism === {{:Definition:Kernel of Magma Homomorphism}} === Kernel of Group Homomorphism === {{:Definition:Kernel of Group Homomorphism}} === Kernel of Ring Homomorphism === {{:Definition:Kernel of Ring Homomorphism}} === Kernel of Linear Transformation === {{:Definition:Kernel of Linear Transformation}} === Kernel of Homomorphism of Differential Complexes === {{:Definition:Kernel of Homomorphism of Differential Complexes}}
Definition:Entropic Structure
An '''entropic structure''' is an algebraic structure $\struct {S, \circ}$ such that $\circ$ is an entropic operation.
Definition:Ringoid (Abstract Algebra)
A '''ringoid''' is a triple $\struct {S, *, \circ}$ where: :$S$ is a set :$*$ and $\circ$ are binary operations on $S$ :the operation $\circ$ distributes over $*$. That is: :$\forall a, b, c \in S: a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}$ :$\forall a, b, c \in S: \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}$
Definition:Semiring (Abstract Algebra)
A '''semiring''' is a ringoid $\struct {S, *, \circ}$ in which: :$(1): \quad \struct {S, *}$ forms a semigroup :$(2): \quad \struct {S, \circ}$ forms a semigroup. That is, such that $\struct {S, *, \circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in S         | m = a * b \in S         | c = Closure under $*$ }} {{axiom | n = \text A 1         | q = \forall a, b, c \in S         | m = \paren {a * b} * c = a * \paren {b * c}         | c = Associativity of $*$ }} {{axiom | n = \text M 0         | q = \forall a, b \in S         | m = a \circ b \in S         | c = Closure under $\circ$ }} {{axiom | n = \text M 1         | q = \forall a, b, c \in S         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | c = Associativity of $\circ$ }} {{axiom | n = \text D         | q = \forall a, b, c \in S         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c}         | c = $\circ$ is distributive over $*$ }} {{end-axiom}} These are called the '''semiring axioms'''.
Definition:Cancellable Semiring
A '''cancellable semiring''' is a semiring $\left({S, *, \circ}\right)$ in which all the elements of $S$ are cancellable for $*$.
Definition:Ring (Abstract Algebra)
A '''ring''' $\struct {R, *, \circ}$ is a semiring in which $\struct {R, *}$ forms an abelian group. That is, in addition to $\struct {R, *}$ being closed, associative and commutative under $*$, it also has an identity, and each element has an inverse.
Definition:Ring Zero
Let $\struct {R, +, \circ}$ be a ring. The identity for ring addition is called the '''ring zero''' (of $\struct {R, +, \circ}$). It is denoted $0_R$ (or just $0$ if there is no danger of ambiguity).
Definition:Ring Negative
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $x \in R$. The inverse of $x$ with respect to the addition operation $+$ in the additive group $\struct {R, +}$ of $R$ is referred to as the '''(ring) negative''' of $x$ and is denoted $-x$. That is, the '''(ring) negative''' of $x$ is the element $-x$ of $R$ such that: :$x + \paren {-x} = 0_R$
Definition:Commutative Ring
A '''commutative ring''' is a ring $\struct {R, +, \circ}$ in which the ring product $\circ$ is commutative.
Definition:Unity (Abstract Algebra)/Ring
Let $\struct {R, +, \circ}$ be a ring. If the semigroup $\struct {R, \circ}$ has an identity, this identity is referred to as '''the unity of the ring $\struct {R, +, \circ}$'''. It is (usually) denoted $1_R$, where the subscript denotes the particular ring to which $1_R$ belongs (or often $1$ if there is no danger of ambiguity).
Definition:Trivial Ring
A ring $\struct {R, +, \circ}$ is a '''trivial ring''' : :$\forall x, y \in R: x \circ y = 0_R$
Definition:Null Ring
A ring with one element is called '''the null ring'''. That is, the '''null ring''' is $\struct {\set {0_R}, +, \circ}$, where ring addition and the ring product are defined as: :$0_R + 0_R = 0_R$ :$0_R \circ 0_R = 0_R$
Definition:Ring with Unity
Let $\struct {R, +, \circ}$ be a non-null ring. Then $\struct {R, +, \circ}$ is a '''ring with unity'''  the multiplicative semigroup $\struct {R, \circ}$ has an identity element. Such an identity element is known as a unity.
Definition:Unit of Ring
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. === Definition 1 === {{:Definition:Unit of Ring/Definition 1}} === Definition 2 === {{:Definition:Unit of Ring/Definition 2}}
Definition:Group of Units
=== Group of Units of Monoid === {{:Definition:Group of Units/Monoid}} === Group of Units of Ring === {{:Definition:Group of Units/Ring}}
Definition:Division Product
Let $\struct {R, +, \circ}$ be a commutative ring with unity. Let $\struct {U_R, \circ}$ be the group of units of $\struct {R, +, \circ}$. Then we define the following notation: :$\forall x \in U_R, y \in R$, we have: ::$\dfrac y x := y \circ \paren {x^{-1} } = \paren {x^{-1} } \circ y$ $\dfrac y x$ is a '''division product''', and $\dfrac y x$ is voiced '''$y$ divided by $x$'''. We also write (out of space considerations) $y / x$ for $\dfrac y x$. This notation is usually used when $\struct {R, +, \circ}$ is a field.
Definition:Proper Element of Ring
A non-zero element of a ring which does not have a product inverse is called a '''proper element'''.
Definition:Zero Divisor
=== Rings === {{:Definition:Zero Divisor/Ring}} === Commutative Rings === The definition is usually made when the ring in question is commutative: {{:Definition:Zero Divisor/Commutative Ring}} === Algebras === {{:Definition:Zero Divisor/Algebra}}
Definition:Division Ring
A '''division ring''' is a ring with unity $\struct {R, +, \circ}$ with the following properties: === Definition 1 === {{:Definition:Division Ring/Definition 1}} === Definition 2 === {{:Definition:Division Ring/Definition 2}} === Definition 3 === {{:Definition:Division Ring/Definition 3}}
Definition:Field (Abstract Algebra)
A '''field''' is a non-trivial division ring whose ring product is commutative. Thus, let $\struct {F, +, \times}$ be an algebraic structure. Then $\struct {F, +, \times}$ is a '''field''' : :$(1): \quad$ the algebraic structure $\struct {F, +}$ is an abelian group :$(2): \quad$ the algebraic structure $\struct {F^*, \times}$ is an abelian group where $F^* = F \setminus \set 0$ :$(3): \quad$ the operation $\times$ distributes over $+$.
Definition:Skew Field
A '''skew field''' is a division ring whose ring product is specifically '''not''' commutative.
Definition:Integral Domain
=== Definition 1 === {{:Definition:Integral Domain/Definition 1}} === Definition 2 === {{:Definition:Integral Domain/Definition 2}}
Definition:Subring
Let $\struct {R, +, \circ}$ be an algebraic structure with two operations. A '''subring of $\struct {R, +, \circ}$''' is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a ring.
Definition:Subdomain
Let $\left({R, +, \circ}\right)$ be an algebraic structure with two operations. A '''subdomain of $\left({R, +, \circ}\right)$''' is a subset $S$ of $R$ such that $\left({S, +_S, \circ_S}\right)$ is an integral domain.
Definition:Division Subring
Let $\struct {R, +, \circ}$ be a ring. A '''division subring''' of $\struct {R, +, \circ}$ is a subset $S$ of $R$ such that $\struct {S, +_S, \circ_S}$ is a division ring.
Definition:Centralizer
=== Centralizer of a Group Element === {{:Definition:Centralizer/Group Element}} === Centralizer of a Subset of a Group === {{:Definition:Centralizer/Group Subset}} === Centralizer of a Subgroup === {{:Definition:Centralizer/Subgroup}} === Centralizer of a Ring Subset === {{:Definition:Centralizer/Ring Subset}}
Definition:Ideal of Ring
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$. Then $J$ is an '''ideal of $R$''' : :$\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$ that is, : :$\forall r \in R: J \circ r \subseteq J \land r \circ J \subseteq J$
Definition:Congruence Modulo an Ideal
Let $\left({R, +, \circ}\right)$ be a ring, and let $J$ be an ideal of $R$. The notation: :$a \equiv b \pmod J$ is used to mean: :$a + \left({-b}\right) \in J$
Definition:Quotient Ring
Let $\struct {R, +, \circ}$ be a ring. Let $J$ be an ideal of $R$. Let $R / J$ be the (left) coset space of $R$ modulo $J$ with respect to $+$. Define an operation $+$ on $R / J$ by: :$\forall x, y: \paren {x + J} + \paren {y + J} := \paren {x + y} + J$ Also, define the operation $\circ$ on $R / J$ by: :$\forall x, y: \paren {x + J} \circ \paren {y + J} := \paren {x \circ y} + J$ The algebraic structure $\struct {R / J, +, \circ}$ is called the '''quotient ring of $R$ by $J$'''.
Definition:Divisor (Algebra)
=== Ring with Unity === {{:Definition:Divisor (Algebra)/Ring with Unity}} === Natural Numbers === {{:Definition:Divisor (Algebra)/Natural Numbers}} === Integers === As the set of integers form an integral domain, the concept '''divides''' is fully applicable to the integers. {{:Definition:Divisor (Algebra)/Integer}} === Gaussian Integers === As the set of Gaussian integers form an integral domain, the concept '''divides''' is also fully applicable to the Gaussian integers. {{:Definition:Divisor (Algebra)/Gaussian Integer}} === Real Numbers === The concept of '''divisibility''' can also be applied to the real numbers $\R$. {{:Definition:Divisor (Algebra)/Real Number}}
Definition:Multiple
=== Integral Domain === {{:Definition:Multiple/Integral Domain}} === Integers === As the set of integers form an integral domain, the concept of being a '''multiple''' is fully applicable to the integers. {{:Definition:Multiple/Integer}}
Definition:Associate
{{transclude:Definition:Associate/Integral Domain   |title = Integral Domain   |section = def   |link = true   |increase = 1   |header = 3 }} === Integers === As the integers form an integral domain, the definition can be applied directly to the set of integers $\Z$: {{:Definition:Associate/Integers}} === Commutative and Unitary Ring === The concept of associatehood can also be applied to the general commutative and unitary ring, even though there may be (proper) zero divisors in the latter: {{:Definition:Associate/Commutative and Unitary Ring}}
Definition:Trivial Factorization
Let $\struct {D, +, \circ}$ be an integral domain. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. A factorization in $\struct {D, +, \circ}$ of the form $x = u \circ y$, where $u \in U_D$ (that is, where $x$ is an associate of $y$) is called a '''trivial factorization'''. === Non-Trivial Factorization === {{:Definition:Trivial Factorization/Non-Trivial Factorization}}
Definition:Irreducible Element of Ring
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Let $x \in D: x \notin U_D, x \ne 0_D$, that is, $x$ is non-zero and not a unit. === Definition 1 === {{:Definition:Irreducible Element of Ring/Definition 1}} === Definition 2 === {{:Definition:Irreducible Element of Ring/Definition 2}}
Definition:Tidy Factorization
Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Any factorization of $x \in D$ can always be '''tidied''' into the form: :$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$ where $u \in \struct {U_D, \circ}$, and may be $1_D$, and $y_1, y_2, \ldots, y_n$ are all non-zero and non-units. This is done by forming the ring product of all units of a factorization into one unit, and rearranging all the remaining factors as necessary. Such a factorization is called '''tidy'''.
Definition:Equivalent Factorizations
Let $\struct {D, +, \circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. Let there be two tidy factorizations of $x$: :$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m$ :$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n$ These two factorizations are '''equivalent''' if one of the following equivalent statements holds:   :$(1): \quad$ There exists a bijection $\pi: \set {1, \ldots, m} \to \set {1, \ldots, n}$ such that $y_i$ and $z_{\map \pi i}$ are associates of each other for each $i \in \set {1, \ldots, m}$. :$(2): \quad$ The multisets of principal ideals $\multiset {\ideal {y_i}: i = 1, \ldots, m}$ and $\multiset {\ideal {z_i}: i = 1, \ldots, n}$ are equal. The equivalence of the definitions is shown by part $(3)$ of Principal Ideals in Integral Domain.
Definition:Complete Factorization
Let $\struct {D, +, \circ}$ be an integral domain. Let $x$ be a non-zero non-unit element of $D$. A '''complete factorization''' of $x$ in $D$ is a tidy factorization: :$x = u \circ y_1 \circ y_2 \circ \cdots \circ y_n$ such that: :$u$ is a unit of $D$ :all of $y_1, y_2, \ldots, y_n$ are irreducible in $D$.
Definition:Unique Factorization Domain
Let $\struct {D, +, \circ}$ be an integral domain. If, for all $x \in D$ such that $x$ is non-zero and not a unit of $D$: :$(1): \quad x$ possesses a complete factorization in $D$ :$(2): \quad$ Any two complete factorizations of $x$ in $D$ are equivalent then $D$ is a '''unique factorization domain'''.
Definition:Subfield
Let $\struct {S, *, \circ}$ be an algebraic structure with $2$ operations. Let $T$ be a subset of $S$ such that $\struct {T, *, \circ}$ is a field. Then $\struct {T, *, \circ}$ is a '''subfield''' of $\struct {S, *, \circ}$.
Definition:Congruence Modulo Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$. We can use $H$ to define relations on $G$ as follows: === Left Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Left Congruence}} === Right Congruence Modulo Subgroup === {{:Definition:Congruence Modulo Subgroup/Right Congruence}}
Definition:Coset
=== Left Coset === {{:Definition:Coset/Left Coset}} === Right Coset === {{:Definition:Coset/Right Coset}}
Definition:Coset Space
Let $G$ be a group, and let $H$ be a subgroup of $G$. === Left Coset Space === {{:Definition:Coset Space/Left Coset Space}} === Right Coset Space === {{:Definition:Coset Space/Right Coset Space}} === Note === If we are (as is usual) concerned at a particular time with only the left or the right coset space, then the superscript is usually dropped and the notation $G / H$ is used for both the left and right coset space. If, in addition, $H$ is a normal subgroup of $G$, then $G / H^l = G / H^r$ and the notation $G / H$ is then unambiguous anyway.
Definition:Quotient Group
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then the left coset space $G / N$ is a group, where the group operation is defined as: :$\paren {a N} \paren {b N} = \paren {a b} N$ $G / N$ is called the '''quotient group of $G$ by $N$'''.
Definition:Quotient Epimorphism
=== Group === {{:Definition:Quotient Epimorphism/Group}} === Ring === {{:Definition:Quotient Epimorphism/Ring}}
Definition:Central Subgroup
Let $G$ be a group. Let $H$ be a subgroup of $G$ which is a subset of the center of $G$. Then $H$ is a '''central subgroup of $G$'''.
Definition:Normalizer
Let $G$ be a group. Let $S$ be a subset of $G$. Then the '''normalizer of $S$ in $G$''' is the set $\map {N_G} S$ defined as: :$\map {N_G} S := \set {a \in G: S^a = S}$ where $S^a$ is the $G$-conjugate of $S$ by $a$.
Definition:Ordered Ring
Let $\struct {R, +, \circ}$ be a ring. Let $\preceq$ be an ordering compatible with the ring structure of $\struct {R, +, \circ}$. Then $\struct {R, +, \circ, \preceq}$ is an '''ordered ring'''.
Definition:Positive
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''positive'''  $0_R \le x$. The set of all '''positive elements of $R$''' is denoted: :$R_{\ge 0_R} := \set {x \in R: 0_R \le x}$
Definition:Strictly Positive
Let $\struct {R, +, \circ, \le}$ be an ordered ring whose zero is $0_R$. Then $x \in R$ is '''strictly positive'''  $0_R \le x$ and $x \ne 0_R$. One may write (more conveniently) that $0_R < x$ or $x > 0_R$ to express that $x$ is '''strictly positive'''. Thus, the set of all '''strictly positive''' elements of $R$ can be denoted: :$R_{>0_R} := \set {x \in R: x > 0_R}$
Definition:Ordering Compatible with Ring Structure
{{begin-axiom}} {{axiom | n = \text {OR} 1         | lc= $\preccurlyeq$ is compatible with $+$:         | q = \forall a, b, c \in  R         | ml= a \preccurlyeq b         | mo= \implies         | mr= \paren {a + c} \preccurlyeq \paren {b + c} }} {{axiom | n = \text {OR} 2         | lc= Product of Positive Elements is Positive         | q = \forall a, b \in R         | ml= 0_R \preccurlyeq x, 0_R \preccurlyeq y         | mo= \implies         | mr= 0_R \preccurlyeq x \circ y }} {{end-axiom}}
Definition:Ring of Endomorphisms
Let $\struct {G, \oplus}$ be an abelian group. Let $\mathbb G$ be the set of all group endomorphisms of $\struct {G, \oplus}$. Let $*: \mathbb G \times \mathbb G \to \mathbb G$ be the operation defined as: :$\forall u, v \in \mathbb G: u * v = u \circ v$ where $u \circ v$ is defined as composition of mappings. Then $\struct {\mathbb G, \oplus, *}$ is called the '''ring of endomorphisms''' of the abelian group $\struct {G, \oplus}$.
Definition:Field of Quotients
Let $D$ be an integral domain. Let $F$ be a field. === Definition 1 === {{:Definition:Field of Quotients/Definition 1}} === Definition 2 === {{:Definition:Field of Quotients/Definition 2}} === Definition 3 === {{:Definition:Field of Quotients/Definition 3}} === Definition 4 === {{:Definition:Field of Quotients/Definition 4}}
Definition:Ordered Field
Let $\left({R, +, \circ, \preceq}\right)$ be an ordered ring. Let $\left({R, +, \circ}\right)$ be a field. Then $\left({R, +, \circ, \preceq}\right)$ is an '''ordered field'''.
Definition:Ordered Structure
An '''ordered structure''' $\left({S, \circ, \preceq}\right)$ is a set $S$ such that: : $(1): \quad \left({S, \circ}\right)$ is an algebraic structure : $(2): \quad \left({S, \preceq}\right)$ is an ordered set : $(3): \quad \preceq$ is compatible with $\circ$.
Definition:Zero (Number)
The number '''zero''' is defined as being the cardinal of the empty set.
Definition:Multiplication
'''Multiplication''' is the basic operation $\times$ everyone is familiar with. For example: :$3 \times 4 = 12$ :$13 \cdotp 2 \times 7 \cdotp 7 = 101 \cdotp 64$
Definition:Subtraction
=== Natural Numbers === {{:Definition:Subtraction/Natural Numbers}} === Integers === {{:Definition:Subtraction/Integers}} === Rational Numbers === {{:Definition:Subtraction/Rational Numbers}} === Real Numbers === {{:Definition:Subtraction/Real Numbers}} === Complex Numbers === {{:Definition:Subtraction/Complex Numbers}} === Extended Real Subtraction === {{:Definition:Extended Real Subtraction}}
Definition:Finite
=== Finite Cardinal === {{:Definition:Finite Cardinal}} === Finite Set === {{:Definition:Finite Set}} === Finite Extended Real Number === {{:Definition:Finite Extended Real Number}}
Definition:Infinite Set
A set which is not finite is called '''infinite'''. That is, it is a set for which there is no bijection between it and any $\N_n$, where $\N_n$ is the the set of all elements of $n$ less than $n$, no matter how big we make $n$.
Definition:Cardinality
Two sets (either '''finite''' or '''infinite''') which are '''equivalent''' are said to have the same '''cardinality'''. The '''cardinality''' of a set $S$ is written $\card S$.
Definition:Countable Set
Let $S$ be a set. === Definition 1 === {{:Definition:Countable Set/Definition 1}} === Definition 2 === {{:Definition:Countable Set/Definition 2}} === Definition 3 === {{:Definition:Countable Set/Definition 3}}
Definition:Uncountable Set
A set which is infinite but not countable is described as '''uncountable'''.
Definition:Ordered Tuple
Let $n \in \N$ be a natural number. Let $\N^*_n$ be the first $n$ non-zero natural numbers: :$\N^*_n := \set {1, 2, \ldots, n}$
Definition:Operation
An '''operation''' is an object, identified by a symbol, which can be interpreted as a process which, from a number of objects, creates a new object.
Definition:Iterated Binary Operation
=== Indexed Iteration === {{Definition:Indexed Iterated Binary Operation}} === Iteration over Finite Set === {{:Definition:Iterated Binary Operation over Finite Set}} === Iteration over Set with Finite Support === {{Definition:Iterated Binary Operation over Set with Finite Support}}
Definition:Summation
Let $\struct {S, +}$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers. Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Summation/Indexed}} === Definition by Inequality === {{:Definition:Summation/Inequality}} === Definition by Propositional Function === {{:Definition:Summation/Propositional Function}}
Definition:Product Notation (Algebra)
Let $\struct {S, \times}$ be an algebraic structure where the operation $\times$ is an operation derived from, or arising from, the multiplication operation on the natural numbers. Let $\tuple {a_1, a_2, \ldots, a_n} \in S^n$ be an ordered $n$-tuple in $S$. === Definition by Index === {{:Definition:Product Notation (Algebra)/Index}}
Definition:Addition
'''Addition''' is the basic operation $+$ everyone is familiar with. For example: :$2 + 3 = 5$ :$47 \cdotp 3 + 191\cdotp 4 = 238 \cdotp 7$
Definition:Power of Element
=== Magma === {{:Definition:Power of Element/Magma}} === Magma with Identity === {{:Definition:Power of Element/Magma with Identity}} === Semigroup === {{:Definition:Power of Element/Semigroup}} === Monoid === {{:Definition:Power of Element/Monoid}} === Group === {{:Definition:Power of Element/Group}} === Ring === {{:Definition:Power of Element/Ring}}
Definition:Indexing Set
Let $I$ and $S$ be sets. Let $x: I \to S$ be a mapping. Let $x_i$ denote the image of an element $i \in I$ of the domain $I$ of $x$. Let $\family {x_i}_{i \mathop \in I}$ denote the set of the images of all the element $i \in I$ under $x$. When a mapping is used in this context, the domain $I$ of $x$ is called the '''indexing set''' of the terms $\family {x_i}_{i \mathop \in I}$. === Index === {{:Definition:Indexing Set/Index}} === Indexed Set === {{:Definition:Indexing Set/Indexed Set}} === Indexing Function === {{:Definition:Indexing Set/Function}} === Family === {{:Definition:Indexing Set/Family}} === Term === {{:Definition:Indexing Set/Term}} === Family of Distinct Elements === {{:Definition:Indexing Set/Family of Distinct Elements}} === Family of Sets === {{:Definition:Indexing Set/Family of Sets}} === Family of Subsets === {{:Definition:Indexing Set/Family of Subsets}}
Definition:Integer
The numbers $\set {\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots}$ are called the '''integers'''. This set is usually denoted $\Z$. An individual element of $\Z$ is called '''an integer'''.
Definition:Additive Group of Integers
The '''additive group of integers''' $\struct {\Z, +}$ is the set of integers under the operation of addition.
Definition:Rational Number
A number in the form $\dfrac p q$, where both $p$ and $q$ are integers ($q$ non-zero), is called a '''rational number'''. The set of all '''rational numbers''' is usually denoted $\Q$. Thus: :$\Q = \set {\dfrac p q: p \in \Z, q \in \Z_{\ne 0} }$
Definition:Irrational Number
An '''irrational number''' is a real number which is not rational. That is, an '''irrational number''' is one that can not be expressed in the form $\dfrac p q$ such that $p$ and $q$ are both integers. The set of '''irrational numbers''' can therefore be expressed as $\R \setminus \Q$, where: :$\R$ is the set of real numbers :$\Q$ is the set of rational numbers :$\setminus$ denotes set difference.
Definition:Factorial
Let $n \in \Z_{\ge 0}$ be a positive integer. === Definition 1 === {{:Definition:Factorial/Definition 1}} === Definition 2 === {{:Definition:Factorial/Definition 2}}
Definition:Binomial Coefficient
=== Definition 1 === {{:Definition:Binomial Coefficient/Integers/Definition 1}} === Definition 2 === {{:Definition:Binomial Coefficient/Integers/Definition 2}} === Definition 3 === {{:Definition:Binomial Coefficient/Integers/Definition 3}}