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## 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.

## NCERT Solutions class 12 Maths Exercise 1.1

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes textbook solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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