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## NCERT Solutions class 12 Maths Relations and Functions

**1. Let be defined as Find the function such that **

**Ans. **Given:

Now and

**2. Let be defined as if is odd and if is even. Show that is invertible. Find the inverse of Here, W is the set of all whole numbers.**

**Ans. **Given: defined as

Injectivity: Let be any two odd real numbers, then

Again, let be any two even whole numbers, then

Is is even and is odd, then

Also, if odd and is even, then

Hence,

is an injective mapping.

Surjectivity: Let be an arbitrary whole number.

If is an odd number, then there exists an even whole number such that

If is an even number, then there exists an odd whole number such that

Therefore, every W has its pre-image in W.

So, is a surjective. Thus is invertible and exists.

For :

and

Hence,

#### NCERT Solutions class 12 Maths Miscellaneous

**3. If is defined by find **

**Ans. **Given:

=

#### NCERT Solutions class 12 Maths Miscellaneous

**4. Show that the function defined by R is one-one and onto function.**

**Ans. ** is one-one: For any R – {+1}, we have

Therefore, is one-one function.

If is one-one, let R – {1}, then

It is cleat that R for all R – {1}, also

Because

which is not possible.

Thus for each R – {1} there exists R – {1} such that

Therefore is onto function.

#### NCERT Solutions class 12 Maths Miscellaneous

**5. Show that the function given by is injective.**

**Ans. **Let R be such that

Therefore, is one-one function, hence is injective.

#### NCERT Solutions class 12 Maths Miscellaneous

**6. Give examples of two functions and such that is injective but is not injective.**

**(Hint: Consider **** and **** )**

**Ans. **Given: two functions and

Let and

Therefore, is injective but is not injective.

#### NCERT Solutions class 12 Maths Miscellaneous

**7. Give examples of two functions and such that is onto but is not onto.**

**(Hint: Consider **** and **** )**

**Ans. **Let

These are two examples in which is onto but is not onto.

#### NCERT Solutions class 12 Maths Miscellaneous

**8. Given a non empty set X, consider P (X) which is the set of all subsets of X.**

**Define the relation AR in P (X) as follows:**

**For subsets A, B in P (X), ARB if and only if A****B. Is R an equivalence relation on P (X)? Justify your answer.**

**Ans. **(i) A A R is reflexive.

(ii) A B B A R is not commutative.

(iii) If A B, B C, then A C R is transitive.

Therefore, R is not equivalent relation.

**9. Given a non-empty set X, consider the binary operation * : P (X) x P (X) P (X) given by A * B = A B A, B in P (X), where P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation *.**

**Ans. **Let S be a non-empty set and P(S) be its power set. Let any two subsets A and B of S.

A B S

A B P(S)

Therefore, is an binary operation on P(S).

Similarly, if A, B P(S) and A – B P(S), then the intersection of sets and difference of sets are also binary operation on P(S) and A S = A = S A for every subset A of sets

A S = A = S A for all A P(S)

S is the identity element for intersection on P(S).

**10. Find the number of all onto functions from the set {1, 2, 3, ……., } to itself.**

**Ans. **The number of onto functions that can be defined from a finite set A containing elements onto a finite set B containing elements =

#### NCERT Solutions class 12 Maths Miscellaneous

**11. Let S = and T = {1, 2, 3}. Find of the following functions F from S to T, if it exists.**

**(i) F = **

**(ii) F = **

**Ans. **S = and T = {1, 2, 3}

(i) F =

(ii)

F is not one-one function, since element and have the same image 1.

Therefore, F is not one-one function.

#### NCERT Solutions class 12 Maths Miscellaneous

**12. Consider the binary operation * : R x R R and o = R x R R defined as and R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that R, [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.**

**Ans. Part I**: also operation * is commutative.

Now,

And

Here, operation * is not associative.

**Part II**: R

And,

operation is not commutative.

Now and

Here operation is associative.

**Part III**: L.H.S. =

R.H.S. = = L.H.S. Proved.

Now, another distribution law:

L.H.S.

R.H.S.

As L.H.S. R.H.S.

Therefore, the operation does not distribute over.

#### NCERT Solutions class 12 Maths Miscellaneous

**13. Given a non-empty set X, let * : P (X) x P (X) P (X) be defined as A * B = (A – B) (B – A), A, B P (X). Show that the empty set is the identity for the operation * and all the elements A of P (X) are invertible with A**^{-1} = A. (Hint: and )

^{-1}= A. (Hint: and )

**Ans. **For every A P(X), we have

=

And =

is the identity element for the operation * on P(X).

Also A * A = (A – A) (A – A) =

Every element A of P(X) is invertible with = A.

#### NCERT Solutions class 12 Maths Miscellaneous

**14. Define binary operation * on the set {0, 1, 2, 3, 4, 5} as **

**Show that zero is the identity for this operation and each element **** of the set is invertible with **** being the inverse of **** **

**Ans. **A binary operation (or composition) * on a (non-empty) set is a function * : A x A A. We denote by for every ordered pair A x A.

A binary operation on a no-empty set A is a rule that associates with every ordered pair of elements (distinct or equal) of A some unique element of A.

* | 0 | 1 | 2 | 3 | 4 | 5 |

0 | 0 | 1 | 2 | 3 | 4 | 5 |

1 | 1 | 2 | 3 | 4 | 5 | 0 |

2 | 2 | 3 | 4 | 5 | 0 | 1 |

3 | 3 | 4 | 5 | 0 | 1 | 2 |

4 | 4 | 5 | 0 | 1 | 2 | 3 |

5 | 5 | 0 | 1 | 2 | 3 | 4 |

For all A, we have (mod 6) = 0

And and

0 is the identity element for the operation.

Also on 0 = 0 – 0 = 0 *

2 * 1 = 3 = 1 * 2

#### NCERT Solutions class 12 Maths Miscellaneous

**15. Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and be the functions defined by A and A. Are and equal? Justify your answer.**

**(Hint: One may note that two functions **** and **** such that **** **** **** A, are called equal functions).**

**Ans. **When then and

At and

At and

At and

Thus for each A,

Therefore, and are equal function.

#### NCERT Solutions class 12 Maths Miscellaneous

**16. Let A = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is:**

**(A) 1**

**(B) 2**

**(C) 3**

**(D) 4**

**Ans. **It is clear that 1 is reflexive and symmetric but not transitive.

Therefore, option (A) is correct.

**17. Let A = {1, 2, 3}. Then number of equivalence relations containing (1, 2) is:**

**(A) 1**

**(B) 2**

**(C) 3**

**(D) 4**

**Ans. **2

Therefore, option (B) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**18. Let be the Signum Function defined as and be the Greatest Function given by where is greatest integer less than or equal to Then, does and coincide in (0, 1)?**

**Ans. **It is clear that and

Consider which lie on (0, # 1)

Now,

And

in (0, 1]

Therefore, option (B) is correct.

#### NCERT Solutions class 12 Maths Miscellaneous

**19. Number of binary operation on the set are:**

**(A) 10**

**(b) 16**

**(C) 20**

**(D) 8**

**Ans. **A =

A x A =

= 4

Number of subsets = = 16

Hence number of binary operation is 16.

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book 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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