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Question

Let R be the real line. Consider the following subsets of the plane R×R,S=(x,y):y=x+1&0<x<2,T=(x,y):xyisaninteger Which one of the following is true?


A

Both S&T are equivalence relation on R

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B

Sis an equivalence relation on R but T is not

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C

T is an equivalence relation on R but S is not

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D

Neither S nor T is an equivalence relation on R

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Solution

The correct option is C

T is an equivalence relation on R but S is not


Explanation For The Correct Option:

Determining the correct statement

Given relation: S=(x,y):y=x+1&0<x<2

Checking reflexivity:

Let y=x

x=x+10=1

Which is not possible.

(x,x)S

Thus, it is not reflexive .

Hence S is not an equivalence on R.

Given relation: T=(x,y):xyisaninteger

Checking reflexivity:

Let y=x

x-x0

Since 0 is an integer so (x,x)T

Thus it is reflexive.

Checking symmetricity:

Put (x,y)=(y,x)

y-x-x-yT

(y,x)T

Thus, it is symmetric relation.

Checking transitivity:

If (x,y)T, (y,z)T then (x,z)Z

Consider yx=I1&zy=I2

zx=I1+I2=I(say)(x,z)T

Thus, T is transitive.

Hence,T is an equivalence relation.

Hence, option C is the correct answer.


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