### Chapter 1 Relations And Functions – Exercise 1.1 | Set 1

### Question 11. Show that the relation R in the set A of points in a plane given by R ={ (P,Q): distance of the point P from the origin is the same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as center.

**Solution:**

We can see (P, P) ∈ R since, the distance of point P from the origin is always the same as the distance of the same point P from the origin. Therefore, R is reflexive.

Let (P,Q)∈ R.

⇒The distance of point P from the origin is the same as the distance of point Q from the origin.

⇒The distance of point Q from the origin is the same as the distance of point P from the origin.

So, (Q,P) ∈ R. Therefore, R is symmetric.

Let (P,Q), (Q,S) ∈ R.

⇒The distance of point P from the origin is the same as the distance of point Q from the origin and also, the distance of point P from the origin is the same.

⇒ The distance of points P and S from the origin is the same.

⇒(P, S) ∈ R. Therefore, R is transitive.

Therefore, R is an equivalence relation.

The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.

In other words, if O(0 0) is the origin and OP = k, then the set of all points related to P is at the same as k from the origin. Hence, this set of points forms a circle with the centre as the origin and the circle passes through point P.

### 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 equivalence relation. Consider three right angle 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:**

R is reflexive since, every triangle is similar to itself.

If (T

_{1}, T_{2}) ∈ R, then T_{1}is similar to T_{2}. So, (T_{2}, T_{1}) ∈ R. Therefore, R is symmetric.Let (T

_{1}, T_{2}), (T_{2}, T_{3}) ∈ R, then T_{1}is similar to T_{2}and T_{2}is similar to T_{3}. So, T_{1}is also similar to T_{3}. Therefore, (T_{1}, T_{3}) ∈ R so, R is transitive.Thus, R is an equivalence relation.

We observe,

(3/6)=(4/8)=(5/10)=1/2

Therefore, the corresponding sides of triangles T

_{1}and T_{3}are in the same ratio. Then, triangle T_{1}is similar to triangle T_{3}.Hence, T

_{1}is related to T_{3}.

### Question 13. Show that the relation R defined in the set A of all polygons as 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 sides 3, 4 and 5?

**Solution:**

R is reflexive since(P

_{1}, P_{2}) ∈ R as the same polygon has the same number of sides with itself.Let (P

_{1}, P_{2}) ∈ R, then P_{1}and P_{2}have the same number of sides. So, (P_{2}, P_{1}) ∈ R. Therefore, R is symmetric.Let (P

_{1}, P_{2}), (P_{2}, P_{3}) ∈ R, then P_{1}and P_{2}have the same number of sides. Also, P_{2}and P_{3}have the same number of sides. So, P_{1}and P_{3}have the same number of sides ,i.e., (P_{1}, P_{3}) ∈ R. Therefore, R is transitive.Hence, R is an equivalence relation.

The elements in A related to the right-angled triangle (T) with sides 3, 4 and 5 are those polygons which have 3 sides(since, T is a polygon with 3 sides).

Hence, the set of all elements in A related to triangle T is the set of all triangles.

**Question** 14. 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=2x+4.

**Solution:**

R is reflexive as any line L

_{1}is parallel to itself i.e., (L_{1}, L_{2}) ∈ R.Let (L

_{1}, L_{2}) ∈ R, then L_{1}is parallel to L_{2}. So, (L_{2}, L_{1}) ∈ R. Therefore, R is symmetric.Let (L

_{1}, L_{2}), (L_{2}, L_{3}) ∈ R, then L_{1}is parallel to L_{2}and L_{2}is parallel to L_{3}. So, L_{1}is parallel to L_{3}. Therefore, 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 the line 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.

### Question 15. Let R be the relation in the set {1, 2, 3, 4} 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 equivalence relation**.

**Solution:**

It is seen that (a,a) ∈ R, for every a ∈ {1, 2, 3, 4}. Therefore, R is reflexive.

It is seen that (1,2) ∈ R but (2,1) ∉ R. Therefore, R is not symmetric.

Also, it is observed that (a,b), (b,c) ∈ R⇒ (a,c) ∈ R for all a, b, c ∈ {1, 2, 3, 4}. Therefore, R is transitive.

Hence, R is reflexive and transitive but not symmetric. The correct answer is

B.

### 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.

**(A) (2,4) ∈ R**

**(B) (3,8) ∈ R**

**(C) (6,8)**

**(D) (8,7) ∈ R**

**Solution:**

Since b>a, (2,4) ∉ R also, as 3≠8-2, (3,8)∉R and as 8≠7-2. Therefore, (8,7)∉R

Consider (6,8). We have 8>6 and also, 6=8-2. Therefore, (6,8) ∈ R.

The correct answer is

C.

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the **DSA Self Paced Course** at a student-friendly price and become industry ready.