Q. 144.2( 54 Votes )

# Let L be the set of all lines in XY plane and R be the relation in L defined asR = {(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.

It is given that the relation in L defined as

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

R is reflexive as any line L1 is parallel to itself

(L1, L2) ϵ R

Now, Let (L1, L2) ϵ R

L1 is parallel to L2.

L2 is parallel to L1.

(L2, L1) ϵ R

Therefore, R is symmetric.

Now, Let (L1, L2), (L2, L3) ϵ R

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

L1 is parallel to L3.

(L1, L3) ϵ R

Therefore, R is transitive.

Therefore, 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

We know that parallel lines have the same slopes.

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

Therefore, the set of all lines related to the given line by y = 2x + c, where c ϵ R.

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