# Q.14   Let L be the set of all lines in XY plane and R be the relation in L defined as . Show that R is an equivalence relation. Find the set of all lines related to the line

All lines are parallel to itself, so it is reflexive.

Let,

i.e.L1 is parallel to T2.

L1 is parallel to L2 is same as L2 is parallel to L1 i.e.

Hence,it is symmetric.

Let,

and    i.e. L1 is parallel to L2  and L2 is parallel  to L3 .

L1 is parallel to L3   i.e.

Hence, it is transitive,

Thus,   , is equivalence relation.

The set of all lines related to the line  are lines parallel to

Here,  Slope = m = 2      and     constant = c = 4

It is known that slope of parallel lines are equal.

Lines parallel to this (  )  line  are   ,

Hence, set of all parallel lines to  are .

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