Determine which of these relations are transitive. The variables x, y, x', y' represent integers. A. x∼yx∼y if and only if xyxy is positive. B. x∼yx∼y if and only if x+yx+y is positive. C. (x,y)∼(x′,y′)(x,y)∼(x′,y′) if and only if x−y=x′−y′x−y=x′−y′.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 2E: 2. In each of the following parts, a relation is defined on the set of all integers. Determine in...

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Question

Determine which of these relations are transitive. The variables x, y, x', y' represent integers.

A. x∼yx∼y if and only if xyxy is positive.
B. x∼yx∼y if and only if x+yx+y is positive.
C. (x,y)∼(x′,y′)(x,y)∼(x′,y′) if and only if x−y=x′−y′x−y=x′−y′.
D. x∼yx∼y if and only if xyxy is negative.
E. (x,y)∼(x′,y′)(x,y)∼(x′,y′) if and only if xy′=x′yxy′=x′y.
F. x∼yx∼y if and only if x+yx+y is even.

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