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Question

Let R be the real line consider the following subsets of the planeR×R,S={(x,y):y=x+1and0<x<2},T={(x,y):x-yisaninteger} .Which one of the following is true?


A

T is an equivalence relation on R but S is not

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B

Neither S nor T is an equivalence relation on R

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C

Both S and T are equivalence relations on R

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D

S is an equivalence relation on R but T is not

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Solution

The correct option is A

T is an equivalence relation on R but S is not


Explanation for the correct option:

Determining the correct statement:

Step 1: Checking if S is an equivalence relation

Given, S={(x,y):y=x+1and0<x<2}

Therefore, y<3

Now for (y,x)x=y+1

So it is not satisfying 0<x<2

Hence, S is not symmetric so not equivalence.

Step 2: Checking if T is an equivalence relation

Given,T={(x,y):x-yisaninteger}

For (x,x)x-x=0 is an integer

So it is reflexive.

For (y,x)y-x=-(x-y) is always an integer.

So it is symmetric.

If (x,y)R&(y,z)R, then

(xy)is an integer,

(yz) is an integer,

Now adding both we get

xz which is also an integer.

Thus,T is an equivalence relation.

Hence, option (A) is the correct answer.


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