Search results
Results From The WOW.Com Content Network
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.
[note 3] For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers.
A function that takes a single argument as input, such as () =, is called a unary function. A function of two or more variables is considered to have a domain consisting of ordered pairs or tuples of argument values. The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.
A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not. A bijection [d] A function that is injective and surjective.
An illustrative example is the standard 52-card deck. The standard playing card ranks {A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2} form a 13-element set. The card suits {♠, ♥, ♦, ♣} form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible ...
An n-tuple can be formally defined as the image of a function that has the set of the n first natural numbers as its domain. Tuples may be also defined from ordered pairs by a recurrence starting from ordered pairs; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its n th element.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the infinitely descending chains = {} from the Axiom of regularity. The axiom of pairing also allows for the definition of ordered pairs.