NDA 2018 Maths · Mathematics · Sets, Relations and Functions · medium

On the set mathbbZ of integers a relation R is defined by aRb if and only if a-b is divisible by 3. 1. R is an equivalence relation. 2. R has exactly…

Single correct option

On the set

\mathbb{Z}

of integers a relation

R

is defined by

aRb

if and only if

a-b

is divisible by 3. Consider the following statements: 1.

R

is an equivalence relation. 2.

R

has exactly three distinct equivalence classes. Which of the statements given above is/are correct?

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Answer: (C) Both 1 and 2

a-a=0

is divisible by 3 (reflexive); if

3\mid a-b

then

3\mid b-a

(symmetric); if

3\mid a-b

and

3\mid b-c

then

3\mid a-c

(transitive). So statement 1 is true. Equivalence classes are the residue classes mod 3, namely

[0],[1],[2]

, so exactly three. Both are correct.

Original question written in the NDA 2018 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake