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For example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. A finite union is the union of a finite number of sets; the phrase does not imply that the union set is a finite set ...
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".
To investigate the left distributivity of set subtraction over unions or intersections, consider how the sets involved in (both of) De Morgan's laws are all related: () = = () always holds (the equalities on the left and right are De Morgan's laws) but equality is not guaranteed in general (that is, the containment might be strict).
Intersections of the unaccented modern Greek, Latin, and Cyrillic scripts, considering only the shapes of the letters and ignoring their pronunciation Example of an intersection with sets The intersection of two sets A {\displaystyle A} and B , {\displaystyle B,} denoted by A ∩ B {\displaystyle A\cap B} , [ 3 ] is the set of all objects that ...
The simple theorems in the algebra of sets are some of the elementary properties of the algebra of union (infix operator: ∪), intersection (infix operator: ∩), and set complement (postfix ') of sets. These properties assume the existence of at least two sets: a given universal set, denoted U, and the empty set, denoted {}.
The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B . Finally, the relative complement of B relative to A , also known as the set theoretic difference of A and B , is the set of all objects that belong to A but not to B .
If some power set ℘ is partially ordered in the usual way (by ) then joins are unions and meets are intersections; in symbols, = = (where the similarity of these symbols may be used as a mnemonic for remembering that denotes the join/supremum and denotes the meet/infimum [note 1]).
The open sets and closed sets of any topological space are closed under both unions and intersections. [ 1 ] On the real line R , the family of sets consisting of the empty set and all finite unions of half-open intervals of the form ( a , b ] , with a , b ∈ R is a ring in the measure-theoretic sense.