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I have a doubt, kindly clarify. - Sets, Relations and Functions - JEE Main-2

Let R be the set of real numbers.

Statement-1 : A=\left \{ (x,y)\in R\times R:y-x\; is\: an\; integer \right \} is an equivalence relation on R.

Statement-2 : 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.

 
Answers (1)
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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.

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