Question

Let R be the relation on the set of real numbers such that xRy if and only if x and y are real numbers that differ by less than 1, that is, |x − y| < 1. Which of the following pair or pairs can be used as a counterexample to show this relation is not an equivalence relation?

A) (1, 1)
B) (1, 1.8), (1.8, 3)
C) (1, 1), (3, 3)
D) (1, 1), (1, 1.5)

Answer 1

A relation is equivalence relation if it is transitive, reflexive and symmetric

A relation is called reflexive if xRx for all possible values of x.

A relation is called symmetric if xRy then yRx should be possible.

A relation is called transitive if xRy and yRz then x should be related to z.

Here option (b) contains 3 numbers 1,1.8 and 3

It doesn't contains relation like (1, 1) (3, 3)...

This the given relation is not reflexive which means it is not equivalence relation.

Option (d) contains number like 1 and 1.5

It contains (1, 1) but doesn't contain (1.5, 1.5). Thus the given relation is not reflexive which makes it non equivalence relation.

There fore the answer is option (b) and (d)

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