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Concept
List of set identities and relations
Summary
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.
The binary operations of set union () and intersection () satisfy many identities. Several of these identities or "laws" have well established names.
Throughout this article, capital letters (such as and ) will denote sets. On the left hand side of an identity, typically,
will be the eft most set,
will be the iddle set, and
will be the ight most set.
This is to facilitate applying identities to expressions that are complicated or use the same symbols as the identity.
For example, the identity
may be read as:
For sets and define:
and
where the is sometimes denoted by and equals:
One set is said to another set if Sets that do not intersect are said to be .
The power set of is the set of all subsets of and will be denoted by
Universe set and complement notation
The notation
may be used if is a subset of some set that is understood (say from context, or because it is clearly stated what the superset is).
It is emphasized that the definition of depends on context. For instance, had been declared as a subset of with the sets and not necessarily related to each other in any way, then would likely mean instead of
If it is needed then unless indicated otherwise, it should be assumed that denotes the universe set, which means that all sets that are used in the formula are subsets of
In particular, the complement of a set will be denoted by where unless indicated otherwise, it should be assumed that denotes the complement of in (the universe)
Assume
Identity:
Definition: is called a left identity element of a binary operator if for all and it is called a right identity element of if for all A left identity element that is also a right identity element if called an identity element.
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