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7 Fill in the blanks Consider the incomplete proof below: Theorem: Let f: A→A be any function from a set to itself, and define g: A →A by g(x) = f(f(x)). Then f is injective if and only if g is injective. Proof: First, we will prove that assuming that Let x, y € A be arbitrary, and suppose g(x) = g(y), or in other words f(f(x)) = f(f(y)). By the assumption we made, this means that f(x) = f(y), and by using the same assumption again, we get x = y. Second, we will prove that suppose f(x) = f(y). Then assuming that , so g(x) = g(y); by assumption, this means that x = y. Let x, y € A be arbitrary, and Which part of the theorem is being proved in the first paragraph, and which part is being proved in the second paragraph? In the second paragraph, explain how we can go from f(x) = f(y) to g(x) = g(y).

7 Fill in the blanks Consider the incomplete proof below: Theorem: Let f: A→A be any function from a set to itself, and define g: A →A by g(x) = f(f(x)). Then f is injective if and only if g is injective. Proof: First, we will prove that assuming that Let x, y € A be arbitrary, and suppose g(x) = g(y), or in other words f(f(x)) = f(f(y)). By the assumption we made, this means that f(x) = f(y), and by using the same assumption again, we get x = y. Second, we will prove that suppose f(x) = f(y). Then assuming that , so g(x) = g(y); by assumption, this means that x = y. Let x, y € A be arbitrary, and Which part of the theorem is being proved in the first paragraph, and which part is being proved in the second paragraph? In the second paragraph, explain how we can go from f(x) = f(y) to g(x) = g(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 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...
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7 Fill in the blanks Consider the incomplete proof below: Theorem: Let f: A→A be any function from a set to itself, and define g: A →A by g(x) = f(f(x)). Then f is injective if and only if g is injective. Proof: First, we will prove that assuming that Let x, y € A be arbitrary, and suppose g(x) = g(y), or in other words f(f(x)) = f(f(y)). By the assumption we made, this means that f(x) = f(y), and by using the same assumption again, we get x = y. Second, we will prove that suppose f(x) = f(y). Then assuming that , so g(x) = g(y); by assumption, this means that x = y. . Let x, y € A be arbitrary, and Which part of the theorem is being proved in the first paragraph, and which part is being proved in the second paragraph? In the second paragraph, explain how we can go from f(x) = f(y) to g(x) = g(y).
Transcribed Image Text:7 Fill in the blanks Consider the incomplete proof below: Theorem: Let f: A→A be any function from a set to itself, and define g: A →A by g(x) = f(f(x)). Then f is injective if and only if g is injective. Proof: First, we will prove that assuming that Let x, y € A be arbitrary, and suppose g(x) = g(y), or in other words f(f(x)) = f(f(y)). By the assumption we made, this means that f(x) = f(y), and by using the same assumption again, we get x = y. Second, we will prove that suppose f(x) = f(y). Then assuming that , so g(x) = g(y); by assumption, this means that x = y. . Let x, y € A be arbitrary, and Which part of the theorem is being proved in the first paragraph, and which part is being proved in the second paragraph? In the second paragraph, explain how we can go from f(x) = f(y) to g(x) = g(y).
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