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The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, always 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 ...
If one finds this unappealing then one can instead define a function as the ordered pair (,), where is a functional relation and is its codomain, but we do not take this approach in this article (more elegantly, if one first defines ordered triples - for example as (,,) = (, (,)) - then one could define a function as the ordered triple (,,) so ...
The following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] 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.
In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.
Given its domain and its codomain, a function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. [ note 1 ] [ 4 ] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
A relation R containing only one ordered pair is also transitive: if the ordered pair is of the form (,) for some the only such elements ,, are = = =, and indeed in this case , while if the ordered pair is not of the form (,) then there are no such elements ,, and hence is vacuously transitive.
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.