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Question

Which of the following statements is not correct for the relation R defined by aRb, if and only if b lives within one kilometre from a?


A

R is reflexive

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B

R is symmetric

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C

R is not anti-symmetric

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D

None of the above

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Solution

The correct option is C

R is not anti-symmetric


Explanation for correct option:

Anti symmetric: If x,yR where xy, then y,xR

From the symmetric condition

a lives within one kilometer from b

i.e., (a,b)R

b lives within one kilometer from a

i.e., (b,a)R

Since (a,b)R and (b,a)R holds when ab.

So, the anti-symmetric does not hold.

Therefore, the final answer is option (c).

Explanation for incorrect options:

The given condition aRb happens if and only if b lives within one kilometer from a

For option (a)

Reflexive: A relation R defined on a set Ais called reflexive, if a is an element of set A then a,aR.
a lives within one kilometer from a. so,
(a,a)R

i.e., aRa

Therefore R is reflexive.

For option (b)

Symmetric: If x,yR where xy, then y,xR

If, a lives within one kilometer from b

i.e., .(a,b)R

b lives within one kilometer from a

(b,a)R

Therefore R is symmetric.

Hence option (c) is the correct option.


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