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Class 12 NCERT Solutions- Mathematics Part I – Chapter 1 Relations And Functions – Exercise 1.1 | Set 2
  • Last Updated : 05 Apr, 2021

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={(T1 ,T2 ): T1 is similar to T2 }, is equivalence relation. Consider three right angle 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:

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

If (T1, T2) ∈ R, then T1 is similar to T2. So, (T2, T1) ∈ R. Therefore, R is symmetric.

Let (T1, T2), (T2, T3) ∈ R, then T1 is similar to T2 and T2 is similar to T3. So, T1 is also similar to T3. Therefore, (T1, T3) ∈ 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 T1 and T3 are in the same ratio. Then, triangle T1 is similar to triangle T3.

Hence, T1 is related to T3.

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

Solution:

R is reflexive since(P1, P2) ∈ R as the same polygon has the same number of sides with itself.

Let (P1, P2) ∈ R, then P1 and P2 have the same number of sides. So, (P2, P1) ∈ R. Therefore, R is symmetric.

Let (P1, P2), (P2, P3) ∈ R, then P1 and P2 have the same number of sides. Also, P2 and P3 have the same number of sides. So, P1 and P3 have the same number of sides ,i.e., (P1, P3) ∈ 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={(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:

R is reflexive as any line L1 is parallel to itself i.e., (L1, L2) ∈ R.

Let (L1, L2) ∈ R, then L1 is parallel to L2. So, (L2, L1) ∈ R. Therefore, R is symmetric.

Let (L1, L2), (L2, L3) ∈ R, then L1 is parallel to L2 and L2 is parallel to L3. So, L1 is parallel to L3. 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.

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