(d) How many functions f: A→ A are there so that fRIA? 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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Need help with 1(D)
![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.
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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.
:
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