Question

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 = 2 x + 4.

Answer 1

The given relation R in the set L of all lines in XY plane is defined as R={ ( L 1 , L 2 ): L 1  is parallel to  L 2 }.

A line is always parallel to itself.

⇒( L 1 , L 1 )∈R

So R is reflexive.

Let, ( L 1 , L 2 )∈R.

⇒ L 1 is parallel to L 2 .

Then L 2 is parallel to L 1 .

⇒( L 2 , L 1 )∈R

So, Ris symmetric.

Let, ( L 1 , L 2 )and ( L 2 , L 3 )∈R.

⇒ L 1 is parallel to L 2

Also, L 2 is parallel to L 3 .

⇒ L 1 is parallel to L 3 .

So ( L 1 , L 3 )∈R, hence Ris transitive.

Thus R is an equivalence relation.

The equation of line is,

y=2x+4(1)

Compare equation (1) with the standard equation of a line,

y=mx+c(2)

Here m is the slope of the line and c is a constant.

⇒m=2

Since parallel lines have same slope, the line parallel to the given line has slope, m=2.

So, the equation of all lines parallel to the given line is y=2x+c which gives us the set of all elements in L related to the given line.