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Concept# Union (set theory)

Summary

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
A refers to a union of zero () sets and it is by definition equal to the empty set.
For explanation of the symbols used in this article, refer to the table of mathematical symbols.
The union of two sets A and B is the set of elements which are in A, in B, or in both A and B. In set-builder notation,
For example, if A = {1, 3, 5, 7} and B = {1, 2, 4, 6, 7} then A ∪ B = {1, 2, 3, 4, 5, 6, 7}. A more elaborate example (involving two infinite sets) is:
A = {x is an even integer larger than 1}
B = {x is an odd integer larger than 1}
As another example, the number 9 is not contained in the union of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of even numbers {2, 4, 6, 8, 10, ...}, because 9 is neither prime nor even.
Sets cannot have duplicate elements, so the union of the sets {1, 2, 3} and {2, 3, 4} is {1, 2, 3, 4}. Multiple occurrences of identical elements have no effect on the cardinality of a set or its contents.
List of set identities and relations and Algebra of sets
Binary union is an associative operation; that is, for any sets
Thus, the parentheses may be omitted without ambiguity: either of the above can be written as Also, union is commutative, so the sets can be written in any order.
The empty set is an identity element for the operation of union. That is, for any set Also, the union operation is idempotent: All these properties follow from analogous facts about logical disjunction.
Intersection distributes over union
and union distributes over intersection
The power set of a set together with the operations given by union, intersection, and complementation, is a Boolean algebra. In this Boolean algebra, union can be expressed in terms of intersection and complementation by the formula
where the superscript denotes the complement in the universal set
One can take the union of several sets simultaneously.

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Intersection (set theory)

In set theory, the intersection of two sets and denoted by is the set containing all elements of that also belong to or equivalently, all elements of that also belong to Intersection is written using the symbol "" between the terms; that is, in infix notation. For example: The intersection of more than two sets (generalized intersection) can be written as: which is similar to capital-sigma notation. For an explanation of the symbols used in this article, refer to the table of mathematical symbols.

Element (mathematics)

In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. Writing means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example , are subsets of A. Sets can themselves be elements. For example, consider the set . The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set . The elements of a set can be anything. For example, is the set whose elements are the colors , and .

Complement (set theory)

In set theory, the complement of a set A, often denoted by A∁ (or A′), is the set of elements not in A. When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A. The relative complement of A with respect to a set B, also termed the set difference of B and A, written is the set of elements in B that are not in A.

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