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2. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Define a relation R on F as follows: For all f, g EF, fRg for all i A, f(i) ≤ g(i). (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Prove or disprove: For all fEF, there exists g EF so that fRg. (c) Prove or disprove: There exists g E F so that for all f EF, f Rg. Let fEF be the function f = {(1, 3), (2, 3), (3, 3), (4, 1), (5,5)}. (d) How many functions g EF are there so that f Rg. Explain. (e) How many functions g E F are there so that g Rf. Explain.

2. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Define a relation R on F as follows: For all f, g EF, fRg for all i A, f(i) ≤ g(i). (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Prove or disprove: For all fEF, there exists g EF so that fRg. (c) Prove or disprove: There exists g E F so that for all f EF, f Rg. Let fEF be the function f = {(1, 3), (2, 3), (3, 3), (4, 1), (5,5)}. (d) How many functions g EF are there so that f Rg. Explain. (e) How many functions g E F are there so that g Rf. Explain.

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 2E: 2. In each of the following parts, a relation is defined on the set of all integers. Determine in...
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2. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Define a relation R on F as follows: For all f, g F, fRg for all i A, f(i) ≤ g(i). (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Prove or disprove: For all fEF, there exists g EF so that fRg. (c) Prove or disprove: There exists g E F so that for all f EF, f Rg. Let fEF be the function f = {(1,3), (2, 3), (3, 3), (4, 1), (5,5)}. (d) How many functions g EF are there so that f Rg. Explain. (e) How many functions g E F are there so that g Rf. Explain.
Transcribed Image Text:2. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Define a relation R on F as follows: For all f, g F, fRg for all i A, f(i) ≤ g(i). (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Prove or disprove: For all fEF, there exists g EF so that fRg. (c) Prove or disprove: There exists g E F so that for all f EF, f Rg. Let fEF be the function f = {(1,3), (2, 3), (3, 3), (4, 1), (5,5)}. (d) How many functions g EF are there so that f Rg. Explain. (e) How many functions g E F are there so that g Rf. Explain.
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