8 1.1 Basic Concepts of Set Theory The set 0/ = {x : x ≠ 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 ̸ = 0}, the set of rational numbers. • R, the set of real numbers. • Intervals. For a,b ∈ R, a ≤ b, we defne: [a,b]= {x ∈ R : a ≤ x ≤ b}. (a,b)= {x ∈ R : a < x < b}. [a,b)= {x ∈ R : a ≤ x < b}. (a,b]= {x ∈ R : a < x ≤ b}. We call intervals of the form [a,b] closed intervals and intervals of the form (a,b) open intervals. Moreover, we will use the symbols ∞ and −∞ in the following defnitions: [a,∞)= {x ∈ R : a ≤ x}. (−∞,b]= {x ∈ R : x ≤ b}. (a,∞)= {x ∈ R : a < x}. (−∞,b)= {x ∈ R : x < 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}. 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 Ac .
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