Ordered Pair

In mathematics, an ordered pair (a, b) is a pair of mathematical objects. In the ordered pair (a, b), the object a is called the first entry, and the object b the second entry of the pair. Alternatively, the objects are called the first and second coordinates, or the left and right projections of the ordered pair. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. This attribute is opposite of unordered pair's attribute: the unordered pair {a, b} equals the unordered pair {b, a}.

Ordered pairs are also called 2-tuples, 2-dimensional vectors, or sequences of length 2. The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered n-tuples (ordered lists of n objects). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)), i.e., as one pair nested in another.

Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs.

Read more about Ordered Pair:  Generalities, Defining The Ordered Pair Using Set Theory, Category Theory

Other articles related to "ordered pair, pair, ordered pairs":

Binary Operation - Pair and Tuple
... A binary operation, ab, depends on the ordered pair (a, b) and so (ab)c (where the parentheses here mean first operate on the ordered pair (a, b) and then operate on the result of that using the ...
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... There are several equivalent definitions of ordered pair, and most mathematicians do not identify the ordered pair with just one of these definitions (since ...
Axiom Of Regularity - Elementary Implications of Regularity - Simpler Set-theoretic Definition of The Ordered Pair
... The axiom of regularity enables defining the ordered pair (a,b) as {a,{a,b}} ... See ordered pair for specifics ... This definition eliminates one pair of braces from the canonical Kuratowski definition (a,b) = {{a},{a,b}} ...
Ordered Pair - Category Theory
... in a category of sets represents the set of ordered pairs, with the first element coming from A and the second coming from B ...
... The locks-and-keys approach represents pointers as ordered pairs (key, address) where the key is an integer value ... variable's cell and into the variable's ordered pair ... When a variable is deallocated, the key of its ordered pair is modified to hold a value different from the variable's cell ...

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