Let us review some basic elements of set theory:

Logical inference In mathematical logic the word implies means “leads to the logical inference that” and can be represented by the symbol of the double shaft arrow, ⇒. This represents a strong causal relation.

Definition of a set We think of a set as a group or collection, defined by the items in the collection. A typical set might consist of all UCSD freshmen, all surfers in Southern California (there is obviously some overlap here), or the positive integers between 1 and 10. We might call a set by another name, such as a collection, a family, a class, an aggregate, or an ensemble. We use the notation of a pair of braces, { }, to denote a set. We can use a description of elements of the set to define the set. Thus, the entity denoted {x | x has property P} is the set of all things with property P (whatever that is). The set of positive integers between 1 and 10 can be expressed then as {1, 2, …, 9, 10} or, equivalently, as {x | x is an integer, 1 ≤ x ≤ 10}.

Elements of a set The elements of a set are the things in the collection. If x is an element of the set A, we write x ∈ A.