Sets and Set Operations

Abstract

Subsets and predicates. For these notes we'll look at one set U , called the universal set, and its subsets. In later notes, we'll build new sets out of old ones using the product construction and the powerset construction. The universal set U corresponds to the domain of discourse in predicate logic when we're only considering unary predicates on that domain. Let, for instance, U be the set of integers, usually denoted Z. A couple of unary predicates for this domain are S(x): \x is a perfect square," and P (x): \x is a positive integer." These two predicates correspond to two subsets of U . The rst corresponds to the set of perfect squares which includes 0; 1; 4; 9; etc., and the second corresponds to the set of positive integers which includes 1; 2; 3; etc. The subset that corresponds to a unary predicate is called the extent of the predicate. There's such a close correspondence between a unary predicate and it's extent that we might as well use the same symbol for both. So, we can use S for the subset of perfect squares, or S for the predicate which indicates with the notation S(x) whether an integer x is a perfect square or not. There are a couple of ways to use notation to specify a set. One is by listing its elements, at least the rst few, and hoping the reader can understand your intent.