Q. 144.2( 54 Votes )

# Let L be the set of all lines in XY plane and R be the relation in L defined as

R = {(L_{1}, L_{2}): L_{1} is parallel to L_{2}}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Answer :

It is given that the relation in L defined as

R = {(L_{1}, L_{2}): L_{1} is parallel to L_{2}}

R is reflexive as any line L_{1} is parallel to itself

⇒ (L_{1}, L_{2}) ϵ R

Now, Let (L_{1}, L_{2}) ϵ R

⇒ L_{1} is parallel to L_{2}.

⇒ L_{2} is parallel to L_{1}.

⇒ (L_{2}, L_{1}) ϵ R

Therefore, R is symmetric.

Now, Let (L_{1}, L_{2}), (L_{2}, L_{3}) ϵ R

⇒ L_{1} is parallel to L_{2}. Also, L_{2} is parallel to L_{3}.

⇒ L_{1} is parallel to L_{3}.

⇒ (L_{1}, L_{3}) 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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