Is the binary relation defined on the set R reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Is the binary relation C defined on the set P(N) reflexive? Is it symmet- ric? Is it transitive? Is it an equivalence relation? (Recall that P(N) is the set of subsets of N). Define the binary relation on C as follows: 2₁~22 iff z1 = |22|. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (*Hint*) Define the binary relation~ on Z as follows: a b iff |a − b < 4. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation?

Is the binary relation defined on the set R reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (Hint) Is the binary relation C defined on the set P(N) reflexive? Is it symmet- ric? Is it transitive? Is it an equivalence relation? (Recall that P(N) is the set of subsets of N). Define the binary relation on C as follows: 2₁~22 iff z1 = |22|. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation? (Hint) Define the binary relation~ on Z as follows: a b iff |a − b < 4. Is ~ reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...

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Question

Please do Exercise 17.2.14 part ABCD and please show step by step and explain

Hint A: ≤ is not symmetric – you may show this by giving a counterexample.

Hint C: The “Is it transitive?” question amounts to answering the following: Given z1 ∼ z2 and z2 ∼ z3. Is it always true that z1 ∼ z3? If yes, prove it; and if no, give a counterexample.

Exercise 17.2.14. For each of the following, explain your answers.
00
17.2 PARTITIONS AND PROPERTIES OF BINARY RELATIONS 591
(a) Is the binary relation < defined on the set R reflexive? Is it symmetric?
Is it transitive? Is it an equivalence relation? (*Hint*)
(b) Is the binary relation C defined on the set P(N) reflexive? Is it symmet-
ric? Is it transitive? Is it an equivalence relation? (Recall that P(N) is
the set of subsets of N).
(c) Define the binary relation
reflexive? Is it symmetric?
(*Hint*)
on C as follows: 2₁~22 iff z₁ = |22|. Is ~
Is it transitive? Is it an equivalence relation?
(d) Define the binary relation on Z as follows: ab iff la − b < 4. Is ~
reflexive? Is it symmetric? Is it transitive? Is it an equivalence relation?

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