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Question

If r is a relation from R (set of real numbers) to R defined by r={(a,b)|a,bR;ab+3isanirrationalnumber}. Then, the relation r is


A

an equivalence relation

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B

only reflexive

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C

only symmetric

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D

only transitive

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Solution

The correct option is A

an equivalence relation


Explanation for the correct options:

A relation is reflexive if (a,a)R for every aA.

A relation is symmetric if (a,b)R then (b,a)R.

A relation is transitive if (a,b)R and (b,c)R then (a,c)R.

Check for reflexive relation:

The relation r is defined as r={(a,b)|a,bR;ab+3isanirrationalnumber}

if aR then aa+3=3 is an irrational number

Therefore, (a,a)r which implies that relation r is reflexive.

Check for symmetric relation:

Assume (a,b)r that is ab+3 is an irrational number

Also ba+3 is an irrational then (b,a)r also exists.

Therefore, r is symmetric.

Check for transitive relation:

Assume (a,b)r and (b,c)r

Therefore, both ab+3 and ac+3 are irrational

Adding the above two relation

ab+3+bc+3=ac+23 an irrational number.

Therefore, (a,c)r exists, which implies that r is transitive.

Conclude that r being reflexive, symmetric and transitive, then it is an equivalence relation.

Hence, the correct option is (A).


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