Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.1 APPLICATIONS OF LOGIC TO SET THEORY--SOME PROOFS 123 Proceed as in Examples 6 and 7 to show that, for any three sets X, Y, ...

124 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 Using this definition, prove: (a) (a, b) = (c, d) if and only if a = c and b = d ...

4.2 INFINITE UNIONS AND INTERSECTIONS 125 The collection d is also sometimes denoted {Ai),. or (Ai 1 i = 1,2,3,.. .). EXAMPLE 1 ...

126 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 EXAMPLE 2 Consider the collection of intervals % = (J, 1 n E N], where each Jn = ...

4.2 INFINITE UNIONS AND INTERSECTIONS 127 EXAMPLE 3 Prove that if (A, 1 k = 1,2,3,... ) is a decreasing collection of sets, then ...

128 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 (e) (U;P= A,) n B = U,"=, (A, n B) [Generalized distributivitr, recaII Exercise 1 ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 129 remove much of that difficulty. We will now "deliver" on that promise. In this article, wit ...

130 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 Many of the most familiar functions lead always to the Type I limit scenario and ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 131 [namely, f(2) = 81. The problem is that the two values, 7 and 8, are dif- ferent; this f ha ...

132 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 THE EPSILON-DELTA DEFINITION AND ITS RELATIONSHIP TO TYPES I, 11, AND I11 By defi ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 133 which pictures the vertical "6 band" about the line x = a, and the hori- zontal "E band" ab ...

134 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 Figure 4.2 The graph of y = f (x) in Example 2. Solution It is intuitively clear ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 135 Y Figure 4.4 A graphic indication that 1 # lim,,, f (x). If - 6 < x c 0, then f (x) does ...

198 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 Figure 4.5 Not just any E can be used in a proof that L # lim,,, f(x). The E band ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 137 (b) Figure 4.6 We can prow 0 # lim,,, f (x), btrt the specific due of E we use in the proof ...

138 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 Solution This problem generalizes a part of Example 2, in which we dealt with the ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 139 Variations of the arguments used in Examples 2 and 3 are needed to prove that limits do not ...

140 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 particular E anywhere in such a proof. We must begin by letting an arbi- trary po ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 141 What then would be the situation in the limit problem we proposed earlier? In particular, w ...

142 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 ' f (a) is undefined (b) Figure 4.8 The graphs off and g are identical, except fo ...