Basic Elements of Real Analysis Murray H. Protter Springer [...]... customary) is denoted by 1 M-5 Existence of a reciprocal For each number a different from zero there is one and only one number x such that ax 1 This number x is called the reciprocal of a (or the inverse of a) and is denoted by a−1 (or 1/a) Axioms M-1 through M-4 are the parallels of Axioms A-1 through A-4 with addition replaced by multiplication However, M-5 is not the exact analogue of A-5, since the additional... consists of the entire shaded area, and S1 ∩ S2 consists of the doubly shaded area Similarly, we may form the union and intersection of any number of sets When we write S1 ∪ S2 ∪ ∪ Sn for the union S of the n sets S1 , S2 , , Sn , then S consists of all elements each of which is in at least one of the n sets n We also use the notation S i 1 Si as shorthand for the union of n sets The intersection of. .. of these axioms we were able to define the sum and product of three numbers Before describing the process of defining sums and products for more than three elements, we now recall several definitions and give some notations that will be used throughout the book Definitions The set (or collection) of all real numbers is denoted by R1 The set of ordered pairs of real numbers is denoted by R2 , the set of. .. A set S of numbers is said to be inductive if (a) 1 ∈ S and (b) (x + 1) ∈ S whenever x ∈ S Examples of inductive sets are easily found The set of all real numbers is inductive, as is the set of all rationals The set of all integers, positive, zero, and negative, is inductive The collection of real numbers between 0 and 10 is not inductive, since it satisfies (a) but not (b) No finite set of real numbers... “only one” part of Axiom A-5 Axioms of Multiplication M-1 Closure property If a and b are numbers, there is one and only one number, denoted by ab (or a × b or a · b), called their product M-2 Commutative law For every two numbers a and b, the equality ba ab holds M-3 Associative law For all numbers a, b, and c, the equality (ab)c holds a(bc) 4 1 The Real Number System M-4 Existence of a unit a There... It may happen that two sets S1 and S2 have no elements i in common In such a case their intersection is empty, and we use the term empty set for the set that is devoid of members Most often we will deal with sets each of which is specified by some property or properties of its elements For example, we may speak of the set of all even integers or the set of all rational numbers between 0 and 1 We employ... , the set of ordered triples by R3 , and so on A relation from R1 to R1 is a set of ordered pairs of real numbers; that is, a relation from R1 to R1 is a set in R2 The domain of this relation is the set in R1 consisting of all the first elements in the ordered pairs The range of the relation is the set of all the second elements in the ordered pairs Observe that the range is also a set in R1 A function... · i 1 m+n ai  i m+1 Since the set of natural numbers N is identical with the set of positive integers, the fact that a natural number is positive follows from the axiom of inequality and the definition of N Theorem 1.12 (The well-ordering principle) Any nonempty set T of natural numbers contains a smallest element Proof Let k be a member of T We define a set S of natural numbers by the relation S... P ∈ S and say that P is an element of S or that P belongs to S If S1 and S2 are two sets, their union, denoted by S1 ∪ S2 , consists of all objects each of which is in at least one of the two sets The intersection of S1 and S2 , denoted by S1 ∩ S2 , consists of all objects each of which is in both sets Figure 1.5 Schematically, if S1 is the horizontally shaded set of points in Figure 1.5 and S2 is the... and b 0 is the negation of the 6 1 The Real Number System statement “a 0 or b 0 or both.” Thus (i) applies For the second part of (ii), suppose ab 0 Then a 0 and b 0, for if one of them were zero, Theorem 1.3 would apply to give ab 0 Theorem 1.6 (i) If a (ii) If a 0, then a−1 0 and [(a−1 )−1 ] a 0 and b 0, then (a · b)−1 (a−1 ) · (b−1 ) The proof of this theorem is like the proof of Theorem 1.2 with addition . p. 167 of text for explanation.) Mathematics Subject Classification (1991): 2 6-0 1, 2 6-0 6, 26A54 Library of Congress Cataloging-in-Publication Data Protter, Murray H. Basic elements of real analysis. by a −1 (or 1/a). Axioms M- 1 through M- 4 are the parallels of Axioms A-1 through A-4 with addition replaced by multiplication. However, M- 5 is not the exact analogue of A-5, since the additional. Basic Elements of Real Analysis Murray H. Protter Springer