Set Operations (Union, Intersection, Complement)

Math for CS · Axiom Academy

Combining and comparing sets: union, intersection, complement, and more Before performing set operations, we need a universal set U — the "universe" of all elements under consideration. For the examples below, let U = \ 1,2,3,4,5,6,7,8,9,10\ , A = \ 1,2,3,4,5\ , and B = \ 3,4,5,6,7\ . Example: A B = \ 1,2,3,4,5,6,7\ Associative: (A B) C = A (B C) Disjoint Sets: If A B = , we say A and B are disjoint — they share no elements. Example: A B = \ 1,2\ and B A = \ 6,7\ . Notice that A B B A in general — set difference is not commutative . Example: A B = \ 1,2\ \ 6,7\ = \ 1,2,6,7\ Two of the most important identities in set theory (and all of logic): In words: the complement of a union is the intersection of complements, and vice versa.

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