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Let L be the set of all lines in a pla...

Let L be the set of all lines in a plane and R be the relation in L defined as `R={(L_1,""""L_2): (L_1" is perpendicular to L"_2)}` . Show that R is symmetric but neither reflexive nor transitive.

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Clearly ,any line L cannot be perpendicular to itself .
`therefore (L,L) !in R "for any"Lin A.`
So, R is not reflexive .
Again ,Let `(L_(1),L_(2))in R.` then
`(L_(1),L_(2))in Rimplies L_(1)bot L_(2)`
`implies L_(2)bot L_(1)`
`implies (L_(2),L_(1)) in R.`

`therefore ` R is symmetric .
Now,Let `L_(1),L_(2),L_(3)in A ` such that `L_(1) bot L_(2) and L_(2)bot L_(3).`
then ,clearly `L_(1) ` is not perpendicular to `L_(3)`.
thus,`(L_(1),L_(2))inR and (L_(1),L_(3)) in R "but" (L_(1),L_(3)!in R.`
`therefore ` R is not transitive .
hence ,R is symmetric but neither reflexive nor transitive.
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