## NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1 (Ex 1.1)

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 1 Exercise 1.1 (Ex 1.1) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 1 Relations and Functions Exercise 1.1 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

## NCERT Solutions class 12 Maths Relations and Functions

**1. Determine whether each of the following relations are reflexive, symmetric and transitive:**

**(i) Relation R in the set A = {1, 2, 3, ……….. 13, 14} defined as R = .**

**(ii) Relation R in the set N of natural numbers defined as R =.**

**(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = **

**(iv) Relation R in the set Z of all integers defined as R = **

**(v) Relation R in the set A of human beings in a town at a particular time given by:**

**(a) R = **

**(b) R = **

**(c) R = **

**(d) R = **

**(e) R = **

**Ans. (i)** R = in A = {1, 2, 3, 4, 5, 6, ……13, 14}

Clearly R = {(1, 3), (2, 6), (3, 9), (4, 12)}

Since, R, R is not reflexive.

Again R but R R is not symmetric.

Also (1, 3) R and (3, 9) R but (1, 9) R, R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**(ii)** R = in set N of natural numbers.

Clearly R = {(1, 6), (2, 7), (3, 8)}

Now R, R is not reflexive.

Again R but R R is not symmetric.

Also (1, 6) R and (2, 7) R but (1, 7) R, R is not transitive.

##### Therefore, R is neither reflexive, nor symmetric and nor transitive.

**(iii)** R = in A = {1, 2, 3, 4, 5, 6}

Clearly R = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6)

Now i.e., (1, 1), (2, 2) and (3, 3) R R is reflexive.

Again i.e., (1, 2) R but R R is not symmetric.

Also (1, 4) R and (4, 4) R and (1, 4) R, R is transitive.

Therefore, R is reflexive and transitive but not symmetric.

**(iv)** R = in set Z of all integers.

Now i.e., (1, 1) = 1 – 1 = 0 Z R is reflexive.

Again R and R, i.e., and are an integerR is symmetric.

Also Z and Z and

R, R is transitive.

Therefore, R is reflexive, symmetric and transitive.

**(v)** Relation R in the set A of human being in a town at a particular time.

**(a)** R =

Since R, because and work at the same place. R is reflexive.

Now, if R and R, since and work at the same place and and work at the same place. R is symmetric.

Now, if R and R R. R is transitive

Therefore, R is reflexive, symmetric and transitive.

**(b)** R =

Since R, because and live in the same locality. R is reflexive.

Also R R because and live in same locality and and also live in same locality. R is symmetric.

Again R and R R R is transitive.

Therefore, R is reflexive, symmetric and transitive.

**(c)** R =

is not exactly 7 cm taller than , so R R is not reflexive.

Also is exactly 7 cm taller than but is not 7 cm taller than , so R but R R is not symmetric.

##### Now is exactly 7 cm taller than and is exactly 7 cm taller than then it does not imply that is exactly 7 cm taller than R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**(d)** R =

is not wife of , so R R is not reflexive.

Also is wife of but is not wife of , so R but R R is not symmetric.

Also R and R then R R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**(e)** R =

is not father of , so R R is not reflexive.

Also is father of but is not father of ,

so R but R R is not symmetric.

Also R and R then R R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**2. Show that the relation R in the set R of real numbers defined as R = is neither reflexive nor symmetric nor transitive.**

**Ans.** R = , Relation R is defined as the set of real numbers.

**(i) **Whether R, then which is false. R is not reflexive.

**(ii)** Whether , then and , it is false. R is not symmetric.

**(iii) **Now , , which is false. R is not transitive

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**3. Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = is reflexive, symmetric or transitive.**

**Ans.** R = R

Now R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} and

**(i)** When which is false, so R, R is not reflexive.

**(ii)** Whether , then and , false R is not symmetric.

**(iii)** Now if R, R R

**(iv) **Then and which is false. R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

**4. Show that the relation R in R defined as R = , is reflexive and transitive but not symmetric.**

**Ans. (i)** which is true, so R, R is reflexive.

(ii) but R is not symmetric.

(iii) and which is true. R is transitive.

Therefore, R is reflexive and transitive but not symmetric.

#### NCERT Solutions class 12 Maths Exercise 1.1

**5. Check whether the relation R in R defined by R = is reflexive, symmetric or transitive.**

**Ans. (i)** For which is false. R is not reflexive.

**(ii)** For and which is false. R is not symmetric.

**(iii) **For and , which is false. R is not transitive.

Therefore, R is neither reflexive, nor symmetric and nor transitive.

#### NCERT Solutions class 12 Maths Exercise 1.1

**6. Show that the relation in the set A = {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.**

**Ans. **R = {(1, 2), (2, 1)}, so for , (1, 1) R. R is not reflexive.

Also if then RR is symmetric.

Now R and then does not imply R R is not transitive..

Therefore, R is symmetric but neither reflexive nor transitive.

#### NCERT Solutions class 12 Maths Exercise 1.1

**7. Show that the relation R in the set A of all the books in a library of a college, given by R = is an equivalence relation.**

**Ans. **Books and have same number of pages R R is reflexive.

If R R, so R is symmetric.

Now if R, R R R is transitive.

Therefore, R is an equivalence relation.

#### NCERT Solutions class 12 Maths Exercise 1.1

**8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.**

**Ans. **A = {1, 2, 3, 4, 5} and R = , then R = {(1, 3), (1, 5), (3, 5), (2, 4)}

**(a)** For which is even. R is reflexive.

If is even, then is also even. R is symmetric.

Now, if and is even then is also even. R is transitive.

Therefore, R is an equivalence relation.

**(b)** Elements of {1, 3, 5} are related to each other.

Since all are even numbers

Elements of {1, 3, 5} are related to each other.

Similarly elements of (2, 4) are related to each other.

Since an even number, then no element of the set {1, 3, 5} is related to any element of (2, 4).

Hence no element of {1, 3, 5} is related to any element of {2, 4}.

#### NCERT Solutions class 12 Maths Exercise 1.1

**9. Show that each of the relation R in the set A = given by:**

**(i) R = **

**(ii) R = **is an equivalence relation. Find the set of all elements related to 1 in each case.

**Ans. (a) (i) **A = A = {0, 1, 2, 3, ……….., 12}

Now R =

R = {(4, 0), (0, 4), (5, 1), (1, 5), (6, 2), (2, 6), ….. (12, 9), (9, 12), …. (8, 0), (0, 8), ….. (8, 4), (4, 8), …… (12, 12)}

Here, is a multiple of 4.

is a multiple of 4. R is reflexive.

Also we observe that R is symmetric.

And is the multiple of 4. R is transitive.

Hence R is an equivalence relation.

**(ii)** R = and A = {0, 1, 2, 3, ……….., 12}

R = {(0, 0), (1, 1), (2, 2), …….. (12, 12)}

For R is reflexive.

As then R is symmetric.

Also then R is transitive.

Hence R is an equivalence relation.

**(b) **Now set of all elements related to 1 in each case.

**(i) **Required set = {1, 5, 9} (ii) Required set = {1}

#### NCERT Solutions class 12 Maths Exercise 1.1

**10. Give an example of a relation, which is:**

**(i) Symmetric but neither reflexive nor transitive.**

**(ii) Transitive but neither reflexive nor symmetric.**

**(iii) Reflexive and symmetric but not transitive.**

**(iv) Reflexive and transitive but not symmetric.**

**(v) Symmetric and transitive but not reflexive.**

**Ans. (i)** The relation “is perpendicular to” is not perpendicular to

If then , however if and then is not perpendicular to

So it is clear that R “is perpendicular to” is a symmetric but neither reflexive nor transitive.

**(ii)** Relation R =

We know that is false. Also but is false and if , this implies

Therefore, R is transitive, but neither reflexive nor symmetric.

**(iii)** “is friend of” R =

It is clear that is friend of R is reflexive.

Also is friend of and is friend of R is symmetric.

Also if is friend of and is friend of then

cannot be friend of R is not transitive.

Therefore, R is reflexive and symmetric but not transitive.

**(iv) **“is greater or equal to” R =

It is clear that R is reflexive.

And does not imply R is not symmetric.

But , R is transitive.

Therefore, R is reflexive and transitive but not symmetric.

**(v) **“is brother of” R =

It is clear that is not the brother of R is not reflexive.

Also is brother of and is brother of R is symmetric.

Also if is brother of and is brother of then

can be brother of R is transitive.

Therefore, R is symmetric and transitive but not reflexive.

#### NCERT Solutions class 12 Maths Exercise 1.1

**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 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 centre.**

**Ans. ** **Part I**: R = {(P, Q): distance of the point P from the origin is the same as the distance of the point Q from the origin}

Let P and Q and O (0, 0).

OP = OQ = =

Now, For (P, P), OP = OP R is reflexive.

Also OP = OQ and OQ = OP (P, Q) = (Q, P) R R is symmetric.

Also OP = OQ and OQ = OR OP = OQ R is transitive.

Therefore, R is an equivalent relation.

**Part II**: As = = (let) which represents a circle with centre (0, 0) and radius

#### NCERT Solutions class 12 Maths Exercise 1.1

**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, T2 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?

_{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, T2 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?

**Ans.** **Part I**: R = {(T_{1}, T_{2}): T_{1} is similar to T_{2}} and T_{1}, T_{2} are triangle.

We know that each triangle similar to itself and thus (T_{1}, T_{2}) R R is reflexive.

Also two triangles are similar, then T_{1} T_{2} T_{1} T_{2} R is symmetric.

Again, if then T_{1} T_{2} and then T_{2} T_{3} then T_{1} T_{3} R is transitive.

Therefore, R is an equivalent relation.

**Part II**: It is given that T_{1}, T_{2} and T_{3} are 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

Since, two triangles are similar if corresponding sides are proportional.

Therefore,

Therefore, T_{1} and T_{3} are related.

#### NCERT Solutions class 12 Maths Exercise 1.1

**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?

_{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?

**Ans. Part I**: R = {(P_{1}, P_{2}): P_{1} and P_{2} have same number of sides}

**(i) **Consider the element (P_{1}, P_{2}), it shows P_{1} and P_{2} have same number of sides. Therefore, R is reflexive.

**(ii)** If (P_{1}, P_{2}) R then also (P_{2}, P_{1}) R

(P_{1}, P_{2}) = (P_{2}, P_{1}) as P_{1} and P_{2} have same number of sides, therefore, R is symmetric.

**(iii)** If (P_{1}, P_{2}) R and (P_{2}, P_{3}) R then also (P_{1}, P_{3}) R as P_{1}, P_{2} and P_{3} have same number of sides, therefore, R is transitive.

Therefore, R is an equivalent relation.

**Part II**: we know that if 3, 4, 5 are the sides of a triangle, then the triangle is right angled triangle. Therefore, the set A is the set of right angled triangle.

#### NCERT Solutions class 12 Maths Exercise 1.1

**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

_{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

**Ans. Part I**: R = {(L_{1}, L_{2}): L_{1} is parallel to L_{2}}

(i) It is clear that L_{1} L_{1} i.e., (L_{1}, L_{1}) R R is reflexive.

(ii) If L_{1} L_{2} and L_{2} L_{1} then (L_{1}, L_{2}) R R is symmetric.

(iii) If L_{1} L_{2} and L_{2} L_{3} L_{1} L_{3} R is transitive.

Therefore, R is an equivalent relation.

**Part II**: All the lines related to the line and where is a real number.

#### NCERT Solutions class 12 Maths Exercise 1.1

**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 an equivalence relation.**

**Ans.** 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)}

**(a)** (1, 1), (2, 2), (3, 3), (4, 4) R R is reflexive.

**(b) **(1, 2) R but (2, 1) R R is not symmetric.

**(c)** If (1, 3) R and (3, 2) R then (1, 2) R R is transitive.

Therefore, option (B) is correct.

#### NCERT Solutions class 12 Maths Exercise 1.1

**16. Let R be the relation in the set N given by R = Choose the correct answer:**

**(A) (2, 4) R**

**(B) (3, 8) R**

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

**(D) (8, 7) R**

**Ans.** Given:

**(A)** , Here is not true, therefore, this option is incorrect.

**(B)** and 3 = 6, which is false.

Therefore, this option is incorrect.

**(C)** and 6 = 6, which is true.

Therefore, option (C) is correct.

**(D) **and 8 = 5, which is false

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