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2.4. Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of positive real numbers. Without using words of negation, for each statement below write a sentence that expresses its negation. a) For all x € A, there is a b € B such that b > x. b) There is an x € A such that, for all b € B, b > x. c) For all x, y = R, f(x) = f(y) ⇒ x = y.

2.4. Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of positive real numbers. Without using words of negation, for each statement below write a sentence that expresses its negation. a) For all x € A, there is a b € B such that b > x. b) There is an x € A such that, for all b € B, b > x. c) For all x, y = R, f(x) = f(y) ⇒ x = y.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...
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2.4. Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of positive real numbers. Without using words of negation, for each statement below write a sentence that expresses its negation. a) For all x € A, there is a b € B such that b > x. b) There is an x € A such that, for all b € B, b>x. c) For all x, y eR, f(x) = f(y) ⇒ x = y. d) For all be R, there is an x e R such that f(x) = b. e) For all x, y e R and all € € P, there is a & E P such that |x - y] < 8 implies \f(x) = f(y)] < €. f) For all € € P, there is a & € P such that, for all x, y € R, \x − y| < 8 implies |f(x) = f(y)] < €.
Transcribed Image Text:2.4. Let A and B be sets of real numbers, let f be a function from R to R, and let P be the set of positive real numbers. Without using words of negation, for each statement below write a sentence that expresses its negation. a) For all x € A, there is a b € B such that b > x. b) There is an x € A such that, for all b € B, b>x. c) For all x, y eR, f(x) = f(y) ⇒ x = y. d) For all be R, there is an x e R such that f(x) = b. e) For all x, y e R and all € € P, there is a & E P such that |x - y] < 8 implies \f(x) = f(y)] < €. f) For all € € P, there is a & € P such that, for all x, y € R, \x − y| < 8 implies |f(x) = f(y)] < €.
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