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Question 14

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.

Answer

R = {(L1, L2): L1 is parallel to L2}

R is reflexive as any line L1 is parallel to itself i.e., (L1, L1) ∈ R.

Now,

Let (L1, L2) ∈ R.

L1 is parallel to L2.

L2 is parallel to L1.

⇒ (L2, L1) ∈ R

∴ R is symmetric.

Now,

Let (L1, L2), (L2, L3) ∈R.

L1 is parallel to L2. Also, L2 is parallel to L3.

L1 is parallel to L3.

∴R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.

Slope of line y = 2x + 4 is m = 2

It is known that parallel lines have the same slopes.

The line parallel to the given line is of the form y = 2x + c, where cR.

Hence, the set of all lines related to the given line is given by y = 2x + c, where cR.

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