Question 12: Show that the relation R defined in the set A of all triangles as R = {(T_{1}, T_{2}): T_{1} is similar to T_{2}}, is an equivalence relation. Consider three right angled triangles T_{1} with sides 3, 4, 5 : T_{2} with sides 5, 12, 13: and T_{3} with sides 6, 8, 10. Which triangles among T_{1}, T_{2} and T_{3} are related?

**Solution:**

and T_{1} and T_{2} 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

T_{1} with sides 3, 4, 5

T_{2} with sides 5, 12, 13 and

T_{3} with sides 6, 8, 10.

Now, in triangle, T_{1} and T_{3} proportion of sides is

Since, corresponding sides of triangles, T_{1} and T_{3} are proportional, thus, there triangle T_{1} and T_{3} 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 = {(P_{1}, P_{2}) : P_{1} and P_{2} 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 = {(P_{1}, P_{2}) : P_{1} and P_{2} have same number of sides}

Since, P_{1} and P_{2} have same number of sides.

Since number of sides in P_{1} and P_{2} are equal

Thus, R is symmetric.

Since, number of sides in P_{1}, P_{2} and P_{3} 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 = {(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.

**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

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