# Let L be the set of all lines in XY-plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

We have, L is the set of lines.

R = {(L1, L2) : L1 is parallel to L2} be a relation on L

Now,

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) R

Since a line is always parallel to itself.

(L1, L2) R

R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) R, then (b, a) R

Let L1, L2 L and (L1, L2) R

L1 is parallel to L2

L2 is parallel to L1

(L1, L2) R

R is symmetric

Transitive: For Transitivity, we need to prove that-

If (a, b) R and (b, c) R, then (a, c) R

Let L1, L2 and L3 L such that (L1, L2) R and (L2, L3) R

L1 is parallel to L2 and L2 is parallel to L3

L1 is parallel to L3

(L1, L3) R

R is transitive

Since, R is reflexive, symmetric and transitive, so R is an equivalence relation.

And, the set of lines parallel to the line y = 2x + 4 is y = 2x + c For all c R

where R is the set of real numbers.

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