This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 14 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 14 Solution
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}.
The given relation is R = (L1, L2) : L1 is parallel to L2
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
As we know, any line L is parallel to itself.
⇒ (L, L) ∈ R
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
If any line L is parallel to line M, then the line M is also parallel to line L
⇒ If (L, M) ∈ R then (M, L) ∈ R
∴ R is symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
If any line L is parallel to line M and line M is parallel to line N, then line L is parallel to the line N.
⇒ If (L, M) ∈ R and (M, N) ∈ R then (L, N) ∈ R
∴ R is transitive.
∴ R is reflexive, symmetric as well as transitive.
∴ R is an equivalence relation.
Find the set of all lines related to the line {y = 2x + 4}.
Any line parallel to the line {y = 2x + 4} will be related to it.
As we know, for two lines to be parallel in a given plane, they should have the same slope m.
We know that the slope of the line {y = mx + c} is m
⇒ the slope of the line {y = 2x + 4} is 2.
⇒ the set of all the lines that are parallel to the line {y = 2x + 4} will be all the lines that have the slope 2.
⇒ The set of all lines related to the line {y = 2x + 4} will be all the lines that can be represented by the linear equation {y = 2x + k} where k is a constant
