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Suppose there is a set L ,set of lines and there is a Relation R,

R={<L1,L2> ϵ R if L1 || L2   |    L1,L2 ϵ L }.

Relation R is, _______________.

1. Reflexive

2.Symmetric

3.Antisymmetric

4.Asymmetric

5.Transitive.

Explanation in simple words with Example will be appreciated.

Thanks.

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Let us see one by one :

a) L1 is parallel to L1 i.e. itself hence it is a reflexive relation

b) If L1 is parallel to L2 , then L2 is also parallel to L1 , so it is a symmetric relation..

c) If L1 is parallel to L2 , L2 is parallel to L3 , then L1 is also parallel L3 as per the geometry , so the relation is also transitive..

d) For antisymmetric to hold we can think it as in xRy , x = y is allowed but if x != y then yRx is not allowed..But here if L1 R L2 holds then L2 R L1 also holds for set of lines and L1 and L2 are distinct..Hence it is not antisymmetric..

e) For asymmetric to hold not even same ones allowed that is , xRx is also not allowed besides antisymmetric condition..But here L1 R L1 is allowed and we have seen it is not anti symmetric also..Hence it is not asymmetric relation..

by Veteran (102k points)
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Thank you Sir. I was actually Confused with antisymmetric relation. That if L1RL2 is there so, L2RL1 is also there and if L1 and L2 are distinct then it will not Hold for Antisymmetric Relation. Did I get it Right ?