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Questions: 1. Let A = {1,2,3,4,5). Let F be the set of all functions from A to A. Let R be the relation on F defined by: for all f.g F, fRg if and only if f(1) = g(2) or f(2)= g(1). The identity function IA: A→ A is defined by IA (x) = x for every a Є A. (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Is it true that for all functions f F, there is a function g so that fRg? Prove your answer. (c) Is it true that for all functions f F, there is a function g so that (f,g) R? Prove your answer. (d) How many functions f: A→A are there so that fRIA? Explain.

Questions: 1. Let A = {1,2,3,4,5). Let F be the set of all functions from A to A. Let R be the relation on F defined by: for all f.g F, fRg if and only if f(1) = g(2) or f(2)= g(1). The identity function IA: A→ A is defined by IA (x) = x for every a Є A. (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Is it true that for all functions f F, there is a function g so that fRg? Prove your answer. (c) Is it true that for all functions f F, there is a function g so that (f,g) R? Prove your answer. (d) How many functions f: A→A are there so that fRIA? Explain.

Advanced Engineering Mathematics
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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Questions: 1. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Let R be the relation on F defined by: for all f, g = F, fRg if and only if ƒ (1) = g (2) or f (2) = g(1). The identity function IA: A → A is defined by IA (x) = x for every x EA. (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Is it true that for all functions ƒ € F, there is a function g so that fRg? Prove your answer. (c) Is it true that for all functions ƒ € F, there a function g so that (f, g) & R? Prove your answer. (d) How many functions f: A→ A are there so that fRIA? Explain. 3. Let A = {1,2,3,4,5}. Let f : A → A be the function defined by f = {(1,2), (2, 2), (3, 1), (4,1), (5,5)}. Next, we define the function g : P (A) → P (A) by putting g(Ø) = Ø and for non-empty subset V of A, we put g (V) = {f(x) | x € V}. Let R be the relation on P (A) defined by: For any X, Y EP (A), (X, Y) = R if and only if g (X) = g(Y). (a) Prove that R is an equivalence relation on P (A). (b) How many equivalence classes are there? Explain. (c) Find three different elements of [{1,3,5}], the equivalence class of {1,3,5}. (d) How many elements does [{1,3,5}] have? Explain. :
Transcribed Image Text:Questions: 1. Let A = {1, 2, 3, 4, 5). Let F be the set of all functions from A to A. Let R be the relation on F defined by: for all f, g = F, fRg if and only if ƒ (1) = g (2) or f (2) = g(1). The identity function IA: A → A is defined by IA (x) = x for every x EA. (a) Is R reflexive? symmetric? antisymmetric? transitive? Prove your answers. (b) Is it true that for all functions ƒ € F, there is a function g so that fRg? Prove your answer. (c) Is it true that for all functions ƒ € F, there a function g so that (f, g) & R? Prove your answer. (d) How many functions f: A→ A are there so that fRIA? Explain. 3. Let A = {1,2,3,4,5}. Let f : A → A be the function defined by f = {(1,2), (2, 2), (3, 1), (4,1), (5,5)}. Next, we define the function g : P (A) → P (A) by putting g(Ø) = Ø and for non-empty subset V of A, we put g (V) = {f(x) | x € V}. Let R be the relation on P (A) defined by: For any X, Y EP (A), (X, Y) = R if and only if g (X) = g(Y). (a) Prove that R is an equivalence relation on P (A). (b) How many equivalence classes are there? Explain. (c) Find three different elements of [{1,3,5}], the equivalence class of {1,3,5}. (d) How many elements does [{1,3,5}] have? Explain. :
2. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and T = {2, 4, 6,8}. Let R be the relation on P (S), the power set of S, defined by: For all A, B EP (S), (A, B) = R if and only if AUT=BUT. (a) Prove that R is an equivalence relation on P (S). (b) How many equivalence classes are there? Explain. (c) How many elements does [{1, 2}], the equivalence class of {1,2}, have? Explain. (d) How many subsets X € P (S) are there so that |X| = 5 and X is not an element of [{1,2}]? Explain.
Transcribed Image Text:2. Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9} and T = {2, 4, 6,8}. Let R be the relation on P (S), the power set of S, defined by: For all A, B EP (S), (A, B) = R if and only if AUT=BUT. (a) Prove that R is an equivalence relation on P (S). (b) How many equivalence classes are there? Explain. (c) How many elements does [{1, 2}], the equivalence class of {1,2}, have? Explain. (d) How many subsets X € P (S) are there so that |X| = 5 and X is not an element of [{1,2}]? Explain.
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