Sets and functions

Abstract

The concept of a’ set’ is extremely basic and pervades the whole of present day mathematical thought. Any well-defined collection of objects is a set. For instance we have:

the set of all students in your class

the set of all prime numbers

the set whose members are you and my left foot

So long as we have some way of specifying the collection, then we say it is a set. Our last example above has already made use of the notion of ‘membership’. If A is a set, then the members of the collection A are called either the members of A or the elements of A. (With a concept as simple as a set, there is no way to avoid circular definitions: but we all know what is meant.) We write

$$ x \in A $$

to denote that x is an element of A.