Class 12 Maths

# Relation and Function

## NCERT Solution

### Exercise 1.1 Part 6

Question 12: Show that the relation R defined in the set A of all triangles as R = {(T1, T2): T1 is similar to T2}, is an equivalence relation. Consider three right angled triangles T1 with sides 3, 4, 5 : T2 with sides 5, 12, 13: and T3 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?

Solution:

and T1 and T2 are triangles.

Since, every triangle is similar to itself.

Similarly, two triangles are similar

Thus, R shows equivalence relation.

Again, as given in question,

Three right angled triangles

T1 with sides 3, 4, 5

T2 with sides 5, 12, 13 and

T3 with sides 6, 8, 10.

Now, in triangle, T1 and T3 proportion of sides is

Since, corresponding sides of triangles, T1 and T3 are proportional, thus, there triangle T1 and T3 are similar.

Hence, triangles T1 and T3 are related.

Question 13: Show that the relation R defined in the set A of all polygons R = {(P1, P2) : P1 and P2 have same number of sides} is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with side 3, 4 and 5?

Solution: Given, R = {(P1, P2) : P1 and P2 have same number of sides}

Since, P1 and P2 have same number of sides.

Since number of sides in P1 and P2 are equal

Thus, R is symmetric.

Since, number of sides in P1, P2 and P3 are equal

Thus, R is an equivalence relation.

Now, since 3, 4 and 5 are the sides of given triangles T, which is a Pythagoras triplet, thus, given triangle is a right angled triangle.

Thus, the set A is set of right angled triangles.

Question 14: Let L be the set of all lines in XY – plane and R be the relation to 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.

Solution: Here given,

It is clear that

Thus, R is reflexive.

Thus, R is symmetric.

Thus, R is transitive.

Since, R is reflexive, symmetric and transitive, thus, R is equivalence relation.

Now, the set of parallel lines related to the line y = 2x + 4, is y = 2x + C where C is any arbitrary constant.

Question 15: Let R be the relation in the set {1, 2, 3, 4} is given by R={(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

(A) R is reflexive and symmetric but not transitive.

(B) R is reflexive and transitive but not symmetric

(C) R is symmetric and transitive but not reflexive

(D) R is an equivalence relation.

Answer: (B) R is reflexive and transitive but not symmetric

Explanation: Here, A = {1, 2, 3, 4}

R ={(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}

Thus, Option (B) is correct

Question 16: Let R be the relation in the set N given by R= {(a, b) : a = b – 2, b > 6} Choose the correct answer

Explanation: Given, a = b -2, b > 6

In the case of option (A), a = 2, and b = 4

Here, since b < 6 thus, Option (A) is not correct

In the case of option (B)

a = 3, b = 8, which does not satisfy the equation a = b – 2

Thus, option (B) is not correct.

In the case of option (C)

a = 6 and b = 8

This satisfies the equation a = b – 2

Thus, option (C) is correct

Similarly option (D) also not satisfies the equation a = b – 2

Exercise 1

Exemplar Problems