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Question

Let R1 and R2 be two relations defined as follows:
R1={(a,b)R2:a2+b2Q} and
R2={(a,b)R2:a2+b2Q}, where Q is the set of all rational numbers. Then :

A
R2 is transitive but R1 is not transitive.
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B
R1 is transitive but R2 is not transitive.
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C
Neither R1 nor R2 is transitive.
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D
R1 and R2 are both transitive.
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Solution

The correct option is C Neither R1 nor R2 is transitive.
For R1
Let a=1+2,b=12,c=814
aR1ba2+b2=6Q
bR1cb2+c2=3Q
aR1ca2+c2=3+42Q
R1 is not transitive.

For R2
Let a=1+2,b=2,c=12
aR2ba2+b2=5+22Q
bR2cb2+c2=522Q
aR2ca2+c2=6Q
R2 is not transitive.

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