Ordered pair

In mathematics, an ordered pair is written as (a, b) and refers to two objects where the order matters. If a and b are different, then (a, b) is not the same as (b, a). This contrasts with an unordered pair, written {a, b}, which is always equal to {b, a}.

Morse definition

Ordered pairs are also known as 2-tuples, sequences, or sometimes lists (in computer science) of length two. When the entries are scalars, they are occasionally called 2-dimensional vectors, though this is technically an abuse of terminology because an ordered pair does not have to belong to a vector space. The entries of an ordered pair can themselves be ordered pairs, which allows for a recursive definition of ordered n-tuples; for example, the ordered triple (a, b, c) can be defined as (a, (b, c)).

Generalities

In the ordered pair (a, b), the object a is called the first entry, and b is the second entry. They may also be referred to as the first and second components, coordinates, or left and right projections. The defining property of ordered pairs is that (a₁, b₁) equals (a₂, b₂) if and only if a₁ equals a₂ and b₁ equals b₂. The set of all ordered pairs with first entry from set A and second entry from set B is called the Cartesian product A × B. A binary relation between sets A and B is simply a subset of A × B.

The notation (a, b) can also be used for other purposes, such as representing open intervals on the real number line, but context usually clarifies the intended meaning. To avoid confusion, the ordered pair may be written as ⟨a, b⟩, though this notation has other uses as well. The left and right projections of a pair p are often denoted π₁(p) and π₂(p), or πℓ(p) and πr(p). For arbitrary n-tuples, πⁿᵢ(t) commonly indicates the i-th component.

Informal and formal definitions

Some introductory textbooks give an informal definition: for any two objects a and b, the ordered pair (a, b) is simply a notation that specifies them in that order. This is often contrasted with a set of two elements, noting that in a set the elements must be different, but in an ordered pair they may be equal, and changing the order of distinct entries changes the pair. This description is considered unsatisfactory because it relies on an intuitive understanding of order, though it is widely used without harm.