Overview

In this chapter we review set theoretic terminology and notation. Later we discuss mathematical reasoning.

2.1 Sets

The word set as used in mathematics means a collection of objects. It is customary to denote sets by capital letters like M, M ₁, etc., below. At this stage the concept of a set is best illuminated by examples, so we list a few sets and name them for ease of reference.

In these examples we have used the word all to make it doubly clear that, for instance, every rational number greater than 1 belongs to M ₁. But usually, in mathematics, if someone mentions the set of rational numbers greater than 1 he or she means the set of all such numbers, and this is automatically understood. Thus with a similar understanding we would say that M ₂ is the set of positive even integers.

If an object x belongs to the set M, we say that x is an element of M or that x lies in M, and we denote this by writing x ? M. If x is not an element of M we write x M. A set M is defined by specifying which elements lie in M.

One way of describing a set is to list its elements, separated by commas, and enclose them in braces (curly brackets); for example, M ₇={2, 4}, M ₂={2, 4, 6, 8,