Ordered pair
Ordered pair: a fundamental mathematical object with order significance.
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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}. 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)).
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.
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. A more rigorous approach treats the ordered pair as a primitive notion, with its characteristic property taken as an axiom, as done by the Bourbaki group i
- field
- Mathematics
- known_for
- Defining ordered pairs and their characteristic property
- definition
- A pair of objects where order matters; (a,b) equals (c,d) if and only if a=c and b=d
- also_called
- 2-tuples, sequences, lists (in computer science), 2-dimensional vectors (technically an abuse of terminology)
- set_theoretic_definitions
- Wiener's (1914), Hausdorff's (1914), Kuratowski's (1921)
Lore & Background
The ordered pair is defined by its characteristic property: (a1,b1) = (a2,b2) if and only if a1=a2 and b1=b2. This property is all that is required to understand the role of ordered pairs in mathematics. The set of all ordered pairs with first entry in set A and second entry in set B is called the Cartesian product A × B. Binary relations between sets A and B are subsets of A × B.
Several set-theoretic definitions of ordered pairs exist. Norbert Wiener proposed the first in 1914, using nested sets. Felix Hausdorff proposed a definition around the same time using distinct objects 1 and 2. Kazimierz Kuratowski offered the now-accepted definition in 1921: (a,b) = {{a}, {a,b}}. When a=b, this reduces to {{a}}.
The ordered pair can be taken as a primitive notion with an associated axiom, as done by the N. Bourbaki group in its Theory of Sets (1954). Alternatively, it can be defined formally in set theory, allowing existence and the characteristic property to be proven from set theory axioms.
Reader's Guide
Ordered pairs are a cornerstone of mathematical structure. Their defining property—that two ordered pairs are equal only when both entries match in order—distinguishes them from unordered pairs. This property enables the rigorous definition of Cartesian products, which in turn underpin binary relations and functions. The notation (a,b) is also used for open intervals on the real number line, but context usually clarifies the meaning. Variant notations such as ⟨a,b⟩ exist but have other uses.
Set-theoretic definitions of ordered pairs, such as Kuratowski's, allow ordered pairs to be constructed from sets, providing a foundation for mathematics within set theory. These definitions are not unique; Wiener's and Hausdorff's are historical alternatives. The recursive definition of ordered n-tuples using nested ordered pairs shows how ordered pairs generalize to longer sequences. The left and right projections of a pair are denoted π1(p) and π2(p) (or πℓ(p) and πr(p)). In contexts with arbitrary n-tuples, πni(t) denotes the i-th component.
While informal definitions describe ordered pairs intuitively, the characteristic property is sufficient for mathematical work. The formal definitions in set theory satisfy the need for rigorous foundations without relying on intuitive notions of order.
Did You Know?
- The ordered pair (a,b) is different from (b,a) if a and b are different, unlike the unordered pair {a,b}.
- Ordered pairs can be nested to define ordered n-tuples; for example, (a,b,c) can be defined as (a,(b,c)).
- Norbert Wiener proposed the first set-theoretical definition of the ordered pair in 1914.
- Kazimierz Kuratowski's 1921 definition (a,b) = {{a}, {a,b}} is the now-accepted set-theoretic definition.
Frequently Asked Questions
What is an ordered pair?
An ordered pair, written (a, b), is a mathematical object made of two entries whose sequence carries meaning. It is equal to another pair (c, d) if and only if a = c and b = d.
How does an ordered pair differ from an unordered pair?
Swapping the entries of an ordered pair yields a distinct object, so (a, b) ≠ (b, a) whenever a and b differ. An unordered pair {a, b}, by contrast, is always the same set as {b, a}.
What set-theoretic definitions of the ordered pair exist?
Three classical constructions are recognized in the literature: Wiener's (1914), Hausdorff's (1914), and Kuratowski's (1921). Each encodes the pair purely using sets while preserving the defining equality property.
What other names does an ordered pair go by?
It is commonly called a 2-tuple, a sequence of length two, or a list in computer science. When the entries are scalars, people sometimes say '2-dimensional vector,' though that is technically an abuse of vector-space terminology.
Why is the ordered pair fundamental to mathematics?
It supplies the minimal structure needed to encode relations, functions, and higher-dimensional objects, since any n-tuple can be built recursively from ordered pairs. Without it, set theory could not formally express mappings between collections.
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