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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..

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