   Chapter 8.3, Problem 45ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# The following argument claims to prove that the requirement that an equivalence relation be reflexive is redundant. In other words, it claims to show that if a relation is symmetric and transitive, then it is reflexive. Find the mistake in the argument. “Proof: Let Rbe a relation on a set A and suppose R is symmetric and transitive. For any two elements x and y in A, if xRy then yRx since R is symmetric. Thusit follows by transitivity thatx R x, and hence R is reflexive.”

To determine

The mistake in the given argument.

Explanation

Given information:

Proof:

Let R be a relation on a set A and suppose R is symmetric and transitive. For any two elements x and y in A, if x R y then y R x since R is symmetric. But then it follows by transitivity that x R x. Hence R is reflexive.”

Calculation:

It could happen that for some xA there does not exist yA such that x R y. in this case, we cannot rely on the argument provided in the exercise prompt to show that R is transitive.

Consider the set A={0,1,2} and the relation R={

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