Algebra of Sets

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Algebra of Sets

Set algebra is fundamental to all of modern mathematics. Indeed, sets form the very foundation of mathematics. The algebra of sets consists of the operations union, intersection, and complement. Suppose S and T are sets. The union of S and T is written S T and equals the set that contains every element of S and every element of T and no other elements. The intersection of S and T is written S T and is the set that contains only those elements that are contained in both S and T. S and T are said to be disjoint if their intersection is the empty set. The complement of S in T is the set of all elements in T that are not in S. It is sometimes denoted by T - S or T\S. If S is a subset of T then the complement of S in T can be denoted by S^C. The symmetric difference of S and T is the set of all elements that are either in S or T but not in both S and T.

For example, suppose S = {1, 2, 3} and T = {2, 3, 4}. Then, S T = {1, 2, 3, 4}. S T equals {2, 3}. The complement of S in T = {4}. The symmetric difference of S and T is {1, 4}.

The curly brackets {} stand for the words "the set containing" and sometimes "the set of all". For example "S union T = {x in S or T}", means T equals the set of all elements x that are in S or in T. Parentheses (), in this context, mean "the set equal to". For example "(S (T U))" means the set equal to S union the set equal to the intersection of T with U. The Greek letter epsilon, ∈ means 'is in'. For example x ∈ S means x is in S, or x is an element of S. The symbol ⫅ means "is a subset of" or "is contained in", as in S ⫅ T. The symbol ⫆ means "is a superset of" or "contains", as in T ⫆ S.

Here are the rules for how to combine unions, intersections, and complements.

• S (T U) = (S T) (S U).
• S (T U) = (S T) (T U).
• The complement of (S T) = (the complement of S) (the complement of T).
• The complement of (S T) = (the complement of S) (the complement of T).

A Boolean algebra of a set B is the set of all its subsets together with the three operations union, complement and intersection. The study of Boolean algebras is important to the design of computer chips and integrated circuits.