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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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