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Question

Consider the relations R={(x,y)|x,y are real numbers and x=wy for some rational number w} S={(m/n,p/q)|m,n,p and q are integers such that n,q not equal to 0 and qm=pn}. Then


A

R is an equivalence relation but S is not an equivalence relation

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B

Neither R nor S is an equivalence relation

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C

S is an equivalence relation but R is not an equivalence relation

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D

R and S both are equivalence relations

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Solution

The correct option is C

S is an equivalence relation but R is not an equivalence relation


Explanation for the correct options:

Step 1. For the reflexive function, we have

R:x=wya,aR

Then,

a=wa

Then the value of w will be 1

R is a reflexive

Similarly,

S={mn,pqand qm=pn

Also,

mn=pqqn=pm

S is symmetric.

Step 2. Write the conditions for the function to be equivalence:

if (ab,ab)S then S will be reflexive and

If S(ab,cd)and(cd,ab) then S is symmetric for a function to be transitive

S(2,4),S(4,8)

S(a,b),S(4,8)

S(a,c)

S(2,8)

S is transitive

S is an equivalence equation.

Hence, option (3) is the correct answer


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