Introduction to Mathematical Analysis I - Second Edition

8 1.1 BASIC CONCEPTS OF SET THEORY The set /0 = { x : x 6 = x } is called the empty set . This set clearly has no elements. Using Theorem 1.1.1 , it is easy to show that all sets with no elements are equal. Thus, we refer to the empty set. Throughout this book, we will discuss several sets of numbers which should be familiar to the reader: • N = { 1 , 2 , 3 , . . . } , the set of natural numbers or positive integers . • Z = { 0 , 1 , − 1 , 2 , − 2 , . . . } , the set of integers (that is, the natural numbers together with zero and the negative of each natural number). • Q = { m / n : m , n ∈ Z , n 6 = 0 } , the set of rational numbers . • R , the set of real numbers . • Intervals. For a , b ∈ R , we have [ a , b ] = { x ∈ R : a ≤ x ≤ b } , ( a , b ] = { x ∈ R : a < x ≤ b } , [ a , ∞ ) = { x ∈ R : a ≤ x } , ( a , ∞ ) = { x ∈ R : a < x } , and similar definitions for ( a , b ) , [ a , b ) , ( − ∞ , b ] , and ( − ∞ , b ) . We will say more about the symbols ∞ and − ∞ in Section 1.5 . Since the real numbers are central to the study of analysis, we will discuss them in great detail in Sections 1.4 , 1.5 , and 1.6 . For two sets A and B , the union , intersection , difference , and symmetric difference of A and B are given respectively by A ∪ B = { x : x ∈ A or x ∈ B } , A ∩ B = { x : x ∈ A and x ∈ B } , A \ B = { x : x ∈ A and x / ∈ B } , and A ∆ B = ( A \ B ) ∪ ( B \ A ) . If A ∩ B = /0, we say that A and B are disjoint . The difference of A and B is also called the complement of B in A . If X is a universal set , that is, a set containing all the objects under consideration, then the complement of A in X is denoted simply by A c . Theorem 1.1.2 Let A , B , and C be subsets of a universal set X . Then the following hold: (a) A ∪ A c = X ; (b) A ∩ A c = /0; (c) ( A c ) c = A ; (d) (Distributive law) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) ;