Question

Let L be the set of straight lines and a relation R on L is defined as 'x is parallel to y', state whether R is a equivalence relation.​

Answer 1

Answer:

R={(L 1 ,L 2 ):L 1 is parallel to L 2 }R is reflexive as any line L

1 is parallel to itself i.e., (L 1 ,L 1 )∈R

.Now, let (L 1 ,L 2)∈R.⇒L 1 is parallel to L 2⇒L

2 is parallel to L 1⇒(L 2 ,L 1 )∈R

∴R is symmetric.Now, let (L

1 ,L 2 ),(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 .⇒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 c∈R.

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