SETS AND THEIR PROPERTIES

Much (perhaps all) of mathematics is about studying sets of objects with particular properties. Section 1 introduces sets and some related concepts. Section 1.4 briefly discusses cardi-nality and introduces countable and uncountable sets. Section A is about familiar sets of numbers, including the integers, rationals, and real numbers. The properties of these sets of numbers that make them distinct are discussed in detail. 1. SETS A set is any well-specified collection of elements.1 Sets are conventionally denoted by capital letters, and elements of a set are usually denoted by lower case letters. The notation, a 2 A, means that a is a member of the set A. A set can be defined by listing its elements inside braces. For example, A = f4, 5, 6g means that A is a set of three elements with members 4, 5, and 6. The members of a set need not be explicitly listed. Instead, they can be defined by some logical relation. For example, the same set A could be writte