# Let R be the set of real numbers.Statement-1 : $\dpi{100} A=\left \{ (x,y)\in R\times R:y-x\; is\: an\; integer \right \}$ is an equivalence relation on R.Statement-2 : $\dpi{100} B=\left \{ (x,y)\in R\times R:x=\alpha y\; for\; some\; rational\; number\; \alpha \right \}$ is an equivalence relation on R. Option 1) Statement-1 is true, Statement-2 is false. Option 2) Statement-1 is false, Statement-2 is true. Option 3) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1. Option 4) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

S Sabhrant Ambastha

As we learnt in

Equivalence relation -

Any relation which is reflexive, symmetric and transitive is called an equivalence relation

-

A = {y - x  is an integer}

B = {x = d, y}

A = {y - x is an integer for }

(a, a) it is an integer.

(a, b), (b, a) it is an integer

(a, b) (b, c) (a, c) it is an integer so that A is an equivalence relation.

Correct option is 1.

Option 1)

Statement-1 is true, Statement-2 is false.

This is the correct option.

Option 2)

Statement-1 is false, Statement-2 is true.

This is an incorrect option.

Option 3)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

This is an incorrect option.

Option 4)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

This is an incorrect option.

Exams
Articles
Questions