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Question

Let R be the relation on the set R of all real numbers defined by aRb iff |ab|1. Then R is

A
Reflexive and symmetric
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B
Symmetric only
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C
Transitive only
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D
Anti-symmetric only
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Solution

The correct option is A Reflexive and symmetric
Given,
aRb if |ab|1

(i) Now, |aa|=0<1
aRa, aR
Hence, R is reflexive.

(ii) If aRb|ab|1
|ba|1 (using property of modulus)
bRa
Hence, R is symmetric.

(iii) We know that 1R12 and 12R1, but 121
Hence, R is not anti-symmetric.

(iv)1R2 i.e. |12|=1 and 2R3 i.e. |23|=1, but 1R3 is not possible as |13|=2>1
Hence, R is not transitive.

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