Ordered Pair

An ordered pair is a mathematical object consisting of two components in a specified order, usually written $(a,b)$. The object $a$ is its first component and $b$ its second. Its defining property is that two ordered pairs are equal exactly when their corresponding components are equal. Ordered pairs provide the basis for Cartesian products, binary relations, and functions, allowing mathematics to distinguish positions, roles, and input–output relationships. (people.math.binghamton.edu)

Definition and equality

The characteristic equality rule is

Consequently, $(a,b)$ and $(b,a)$ are different unless $a=b$. For example, $(2,5)\neq(5,2)$, whereas $(3,3)$ is unchanged by exchanging its components. Repetition is allowed: an ordered pair always has two component positions, even when both contain the same object. (people.math.binghamton.edu)

This distinguishes an ordered pair from an unordered pair, represented by the set $\{a,b\}$. Sets do not distinguish the order in which their members are written, so $\{a,b\}=\{b,a\}$. Moreover, $\{a,a\}=\{a\}$, a singleton set. Ordered pairs therefore preserve information about position that a simple set of components does not. The alternative notation $\langle a,b\rangle$ is also used. (people.umass.edu)

Construction in set theory

In set theory, ordered pairs can be represented using sets alone. A standard construction, called the Kuratowski ordered pair, defines

The nested structure distinguishes the first component without requiring the outer set itself to have an order. This construction satisfies the characteristic equality rule and therefore implements the abstract notion of an ordered pair. (math.cuhk.edu.hk)

To see how the components are recoverable, let $p=\{\{a\},\{a,b\}\}$. The intersection of its members is $\{a\}$, while their union is $\{a,b\}$. Thus $a$ is uniquely determined; if the union contains a second, distinct element, that element is $b$, and otherwise $b=a$. In particular,

Although this representing set has only one member, it still encodes two equal component positions. These observations establish why the construction retains the required information even when the components coincide. (math.cuhk.edu.hk)

Cartesian products and projections

For sets $A$ and $B$, their Cartesian product is

For example, if $A=\{1,2\}$ and $B=\{x,y\}$, the product contains $(1,x),(1,y),(2,x),(2,y)$. The first component must come from $A$, and the second from $B$. If either factor is the empty set, the product is empty. (people.math.binghamton.edu)

The coordinate projections are the functions $\pi_1(a,b)=a$ and $\pi_2(a,b)=b$. They extract the two components separately. Exchanging components gives a bijection from $A\times B$ to $B\times A$, with inverse given by the same exchange. This correspondence does not mean the two products are identical: the order of their factors determines which component belongs in each position. (people.math.binghamton.edu)

Relations and functions

A binary relation from $A$ to $B$ is a subset $R\subseteq A\times B$. Membership $(a,b)\in R$ says that $a$ bears the specified relation to $b$. For a relation such as “owes money to,” reversing the pair changes who owes whom. Its inverse relation is

The order of components therefore expresses the direction of the relationship, not merely a typographical convention. (osera.cs.grinnell.edu)

A function $f:A\to B$ has a graph

Every member of the domain occurs as a first component with exactly one associated value in the codomain. A general relation may associate one first component with several second components; a function cannot. Different inputs may nevertheless share an output unless the function is injective. (osera.cs.grinnell.edu)

Coordinate geometry and complex numbers

In analytic geometry, a Cartesian coordinate system represents each point in a plane by an ordered pair $(x,y)$ of real numbers. The first coordinate specifies horizontal position and the second vertical position relative to the axes and origin. Thus $(2,5)$ and $(5,2)$ generally describe different points. An ordered pair solves an equation in two variables when substituting its components makes the equation true. (openstax.org)

A complex number $z=a+bi$ can likewise be represented by $(a,b)$. In the complex plane, the horizontal coordinate is the real part and the vertical coordinate is the imaginary part. For example, $-2+3i$ corresponds to $(-2,3)$. This representation assigns distinct mathematical roles to the two real components. (openstax.org)

Tuples and nested pairs

An ordered pair is the two-component case of an ordered tuple. Longer tuples may be constructed recursively; one convention defines $(a,b,c)$ as $((a,b),c)$. Another represents a tuple as a function on the index set $\{1,2,3\}$, assigning each index its corresponding component. Both preserve componentwise equality. (math.cuhk.edu.hk)

Nested products require attention to grouping. The objects $((a,b),c)$ and $(a,(b,c))$ have different structures, belonging respectively to $(A\times B)\times C$ and $A\times(B\times C)$. There is, however, a natural bijection between them that preserves all three components. Mathematical notation commonly suppresses this representational distinction when it does not affect the argument. (cs.yale.edu)