Miscellaneous

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NCERT Solutions class 12 Maths Miscellaneous

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 AB. 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 )

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.