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In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).
A left identity element that is also a right identity element if called an identity element. The empty set is an identity element of binary union and symmetric difference , and it is also a right identity element of set subtraction :
If a subset does not contain the distinguished element, then it is a subset of the set of all non-distinguished elements. By the induction hypothesis, the number of such subsets is 2 n. Finally, the whole list of subsets of our size-(n + 1) set contains 2 n + 2 n = 2 n+1 elements.
Counting the empty set as a subset, a set with elements has a total of subsets, and the theorem holds because > for all non-negative integers. Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for infinite sets also.
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
The binomial theorem is closely related to the power set. A k –elements combination from some set is another name for a k –elements subset, so the number of combinations, denoted as C(n, k) (also called binomial coefficient) is a number of subsets with k elements in a set with n elements; in other words it's the number of sets with k ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Also, 1, 2, and 3 are members (elements) of the set {1, 2, 3}, but are not subsets of it; and in turn, the subsets, such as {1}, are not members of the set {1, 2, 3}.