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Set is the prototype of a concrete category; other categories are concrete if they are "built on" Set in some well-defined way. Every two-element set serves as a subobject classifier in Set. The power object of a set A is given by its power set, and the exponential object of the sets A and B is given by the set of all functions from A to B.
In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. [1] The set X is called the domain of the function [2] and the set Y is called the codomain of the function. [3] Functions were originally the idealization of how a varying quantity depends on another quantity.
An arrow from x to y indicates that the relation holds between x and y. The relation is represented by the set { (a,a), (a,b), (a,d), (b,a), (b,d), (c,b), (d,c), (d,d) } of ordered pairs. In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1]
The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all the equations or inequalities. If the solution set is empty, then there are no values of the unknowns that satisfy simultaneously all equations and inequalities.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
In set theory, X Y is the notation representing the set of all functions from Y to X. As "2" can be defined as {0, 1} (see, for example, von Neumann ordinals), 2 S (i.e., {0, 1} S) is the set of all functions from S to {0, 1}. As shown above, 2 S and the power set of S, P (S), are considered identical set-theoretically.
Given two sets X and Y, the set of all functions from Y to X is denoted by X Y. Then the following statements hold: If A ~ B and C ~ D then A C ~ B D. A B ∪ C ~ A B × A C for disjoint B and C. (A × B) C ~ A C × B C (A B) C ~ A B×C; These properties are used to justify cardinal exponentiation.
A homomorphism between two algebras A and B is a function h : A → B from the set A to the set B such that, for every operation f A of A and corresponding f B of B (of arity, say, n), h(f A (x 1, ..., x n)) = f B (h(x 1), ..., h(x n)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra the function is from.)