Consider the following two binary relations on the set A=\ a, b, c\ : R_1=\ (c, a)(b, b),(a, c),(c , c),(b, c),(a, a)\ and R_2=\ (a, b),(b, a),(c, c) , (c, a), (a, a), (b, b), (a, c). Then

Correct answer: A. R_2 is symmetric but it is not transitive Both R_1 and R_2 are symmetric as For any (x, y) ∈ R_1 , we have (y, x) ∈ R_1 and similarly for R_2 Now, for R_2,(b, a) ∈ R_2,(a, c) ∈ R_2 but (b, c) R_2 . Similarly, for R_1,(b, c) ∈ R_1,(c, a) ∈ R_1 but (b, a) R_1 . Therefore, neither R_1 nor R_2 is transitive.

JEE Main 201815 AprilMorning ShiftOnline

Exam: JEE Main
Year: 2018
Date: 15 April
Shift: Morning Shift
Mode: Online
Subject: Mathematics
Chapter: Sets and Relations

JEE Main 2018 · 15 April · Morning Shift · Online · Mathematics · Sets and Relations

Consider the following two binary relations on the set A=\ a, b, c\...

Consider the following two binary relations on the set

A=\{a, b, c\}: R_1=\{(\mathrm{c}, a)(b, b),(\mathrm{a}, c),(c

c),(b, c),(a, a)\}

and

\mathrm{R}_2=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{a}),(\mathrm{c}, \mathrm{c})

, (c, a), (a, a), (b, b), (a, c). Then

Options

A $R_2$ is symmetric but it is not transitive
BBoth $R_1$ and $R_2$ are transitive
CBoth $R_1$ and $R_2$ are not symmetric
D $R_1$ is not symmetric but it is transitive

Correct answer

A. $R_2$ is symmetric but it is not transitive

Step-by-step solution

Both

R_1

and

R_2

are symmetric as For any

(x, y) \in R_1

, we have

(y, x) \in R_1

and similarly for

R_2

Now, for

R_2,(b, a) \in R_2,(a, c) \in R_2

but

(b, c) \notin R_2

. Similarly, for

R_1,(b, c) \in R_1,(c, a) \in R_1

but

(b, a) \notin R_1

. Therefore, neither

R_1

nor

R_2

is transitive.

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