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Abhijit Dasgupta Set Theory 2014 With an Introduction to Real Point Sets 10.1007/978-1-4614-8854-5_1

Springer Science+Business Media New York 2014

1. Preliminaries: Sets, Relations, and Functions

Abhijit Dasgupta 1

(1)

Department of Mathematics, University of Detroit Mercy, Detroit, MI, USA

Abhijit Dasgupta

Email:

Abstract

This preliminary chapter informally reviews the prerequisite material for the rest of the book. Here we set up our notational conventions, introduce basic set-theoretic notions including the power set, ordered pairs, Cartesian product, relations, functions, and their properties, sequences, strings and words, indexed and unindexed families, partitions and equivalence relations, and the basic definition of linear order. Much of the material of this chapter can be found in introductory discrete mathematics texts.

1.1 Introduction

Note . In this preliminary chapter, we informally use the familiar number systems N , Z , R , and their properties to provide illustrative examples for sets, relations, and functions. In the next three chapters all of these notions will be formally defined. Thus all our assumptions about these number systems are temporary and will be dropped at the end of this chapter.

We assume basic familiarity with sets and functions, e.g., as found in elementary calculus. Some examples of sets are the real@ R , the set of all real numbersseereal numbers and sets real numbers and sets!intervals real intervals : The open interval ( a , b ) consists of real numbers lying strictly between a and b , and the closed interval [ a , b ] consists of

real numbers x satisfying

a x b . The interval (,) is the entire real line and is denoted by the special symbol R :

In addition we will be using the special symbols N and Z , where

• N consists of the natural numbers starting from 1
  (positive integers).
• Z consists of all integers positive, negative, or zero.

1.1.1 The Principle of Induction

induction!principle of (finite)( We will also assume some familiarity with the principle of induction for the positive integers N . Let P be a property of natural numbers. We will use the notation P ( n ) to stand for the assertion n has the property P . For example, P ( n ) may stand for n ( n 2 + 2) is divisible by 3.

The Principle of Induction.

Let P be a property of natural numbers such that

• P (1) is true.
• For any natural number n , if P ( n ) is true then P ( n + 1) is true.

Then P ( n ) is true for all natural numbers n .

Problem 1.

Show that the principle of induction is equivalent to the principle of strong induction for N which is as follows:

Let P be a property of natural numbers such that

• For any natural number n, if P(m) is true for all natural numbers m < n then P(n) is true.

Then P(n) is true for all natural numbers n.

The natural numbers and the principle of induction will be studied in detail in .induction!principle of (finite))

1.2 Membership, Subsets, and Naive Axioms

Naively speaking, a set A is a collection or group of objects such that membership in A is definitely determined in the sense that given any x , exactly one of x A or x A is true, where the notation

is used to denote that set, sets!membership x is a member of the set A , and the notation

stands for x is not a member of A. For example, we have 3(2,), , 1 N , 0 N , etc.

We say that A is a subset of B , denoted by A B , if every member of A is a member of B . We write A B to denote that A is not a subset of B . A B is also expressed by saying that A is contained in B or B contains A. Thus we have

We are using the symbol as a short-hand for the phrase if and only if (or equivalence of statements). Similarly, the symbol will stand for implication , that is P Q means if P then Q or P implies Q .

We will also often use the abbreviations for for all x , (the universal quantifier ) and for there is some x such that (the existential quantifier ). With such abbreviations, the lines displayed above can be shortened to:

The axiom of!extensionality extensionality!principle of principle of!extensionality principle of extensionality says that two sets having the same members must be identical , that is:

which can also be stated in terms of subsets as:

The axiom of!comprehension!unlimited (naive, unrestricted) comprehension!naive principle of principle of!comprehension, naive naive principle of comprehension is used to form new sets. Given any property P , we write P ( x ) for the assertion x has property P . Then the naive principle of comprehension says that, given any property P , there is a set A consisting precisely of those x for which P(x) is true. In symbols:

We use the qualifier naive to indicate that the principle of comprehension uses the vague notion of property, and unrestricted use of the comprehension principle can cause problems that will be discussed later.

1.2.1 Set Builder Notation

set, sets!notation!set builder( set builder notation( By extensionality, the set A whose existence is given by comprehension from a property P is unique, so we can introduce the set builder notation

to denote the unique set A consisting precisely of those x for which P ( x ) is true, i.e., the set A defined by the condition: x A P ( x ) (for all x ). So,

For example, we have:

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