Exactly one of the three relations below is an equivalence relation. For each of the three: State whether it is the equivalence relation or not. If that relation is the equivalence relation, write a short proof including the correct vo- cabulary for all the properties that an equivalence relation must satisfy. • If not, explain which property of equivalence relations fails to be satisfied. For each of the relations below, the domain is the integers Z. Part A The relation aRb means that a - b < 5. Part B The relation aRb means that a + b is a multiple of 3. Recall that x is a multiple of y if there exists an integer c such that x = cy. Part C The relation aRb means that ab ≥ 0.

Exactly one of the three relations below is an equivalence relation. For each of the three: State whether it is the equivalence relation or not. If that relation is the equivalence relation, write a short proof including the correct vo- cabulary for all the properties that an equivalence relation must satisfy. • If not, explain which property of equivalence relations fails to be satisfied. For each of the relations below, the domain is the integers Z. Part A The relation aRb means that a - b < 5. Part B The relation aRb means that a + b is a multiple of 3. Recall that x is a multiple of y if there exists an integer c such that x = cy. Part C The relation aRb means that ab ≥ 0.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 15E: Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in...

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