Prove or solve the following statements: a. Suppose P is a partition of a set A. Define a relation R on A by declaring xRy if and only if x,y are in X for some X in P. Prove R is an equivalence relation on A. Then prove that P is the set of equivalence classes of R. b. Suppose [a],[b] is in the set of integers Z6 and [a] * [b] = [0]. Is it necessarily true that either [a] = [0] or [b] = [0]? What if [a] and [b] are in the set of integers Z7? c. Suppose f: Z --> Z is defined as f = {(x,4x 5):x is an integer}. State the domain, codomain, and range of f. Find f(10). d. Consider the set f = {(x^3,x):x is a real number}. Is this a function from the set of real numbers (R) to the set of real numbers (R)? Explain.

Prove or solve the following statements: a. Suppose P is a partition of a set A. Define a relation R on A by declaring xRy if and only if x,y are in X for some X in P. Prove R is an equivalence relation on A. Then prove that P is the set of equivalence classes of R. b. Suppose [a],[b] is in the set of integers Z6 and [a] * [b] = [0]. Is it necessarily true that either [a] = [0] or [b] = [0]? What if [a] and [b] are in the set of integers Z7? c. Suppose f: Z --> Z is defined as f = {(x,4x 5):x is an integer}. State the domain, codomain, and range of f. Find f(10). d. Consider the set f = {(x^3,x):x is a real number}. Is this a function from the set of real numbers (R) to the set of real numbers (R)? Explain.

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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