2. Suppose I wish to define a function g : {0, 1, 2, 3, 4}the numbers 0, 1, 2, 3, or 4), and returns an output of the integer with that equivalence class.→ Z that takes an input of an equivalence class mod 5 (so one ofIs g a well-defined function? Why or why not?

Answered: Image /qna-images/answer/8c07a310-bf5c-4150-b98a-f4d6dfab1f25.jpg

Answered: 2. Suppose I wish to define a function…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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Prove that if f is a permutation on A, then (f1)1=f.
4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
9. The definition of an even integer was stated in Section 1.2. Prove or disprove that the set of all even integers is closed with respect to a. addition defined on . b. multiplication defined on .
5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
30. Prove statement of Theorem : for all integers .
6. Prove that if is a permutation on , then is a permutation on .
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.

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2. Suppose I wish to define a function g : {0, 1, 2, 3, 4} → Z that takes an input of an equivalence class mod 5 (so one of the numbers 0, 1, 2, 3, or 4), and returns an output of the integer with that equivalence class. Is g a well-defined function? Why or why not?