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## Ncert Solutions for Class 12

#### Exercise 1.2

1. Show that the function f: R∗→R∗ defined by $$f(x)=\frac{1}{x}$$ is one-one and onto, where R∗ is the set of all non-zero real numbers. Is the result true, if the domain R∗ is replaced by N with co-domain being same as R∗?

2. Check the injectivity and surjectivity of the following functions:

(i) f: N→N given by $$f(x)=x^2$$.

(ii) f: Z→Z given by $$f(x)=x^2$$.

(iii) f: R→R given by $$f(x)=x^2$$.

(iv) f: N→N given by $$f(x)=x^3$$.

(v) f: Z→Z given by $$f(x)=x^3$$.

3. Prove that the Greatest Integer Function f: R→R, given by f(x)=[x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

4. Show that the Modulus Function f: R→R, given by f(x)=|x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.

5. Show that the Signum Function f: R→R, given by:

$$f(x)=\begin{cases}1, \text{if}\:\: x>0\\0, \text{if}\:\: x=0\\1, \text{if}\:\: x<0\end{cases}$$ is neither one-one nor onto.

6. Let A={1,2,3}, B={4,5,6,7} and let f={(1,4),(2,5),(3,6)} be a function from A to B. Show that f is one-one.

7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.

(i) f: R→R defined by f(x)=3–4x.

(ii) f: R→R defined by f(x)=1+x2

8. Let A and B be sets. Show that f: A×B→B×A such that f(a,b)=(b,a) is bijective function.

9. Let f: N→N be defined by

$$f(n)=\begin{cases}\frac{n+1}{2}, \text{if n is odd}\\\frac{n}{2}, \text{if n is even}\end{cases}$$ for all $$n\in N$$.

10. Let A=R–{3} and B=R–{1}. Consider the function f: A→B defined by $$f(x)=(\frac{x-2}{x-3})$$. Is f one-one and onto? Justify your answer.

11 . Let f: R→R be defined as f(x)=x4. Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

12. Let f: R→R be defined as f(x)=3x. Choose the correct answer.

(A) f is one-one onto

(B) f is many-one onto

(C) f is one-one but not onto

(D) f is neither one-one nor onto.

#### Miscellaneous Exercise

1. Show that the function f : R → {x ∈ R : – 1 < x < 1} defined by $$f(x)={x\over {1+|x|}}$$, x∈R is one one and onto function.

2. Show that the function f : R → R given by $$f(x) = x^3$$ is injective.

3. Given a non empty set X, consider P(X) which is the set of all subsets of X.
Define the relation R 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.

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

5. Let A = {– 1, 0, 1, 2}, B = {– 4, – 2, 0, 2} and f, g : A → B be functions defined by $$f (x) = x^2 – x$$, x ∈ A and $$g(x)=2|x-{1\over 2}|-1$$,x∈A. Are f and g equal? Justify your answer.

6. 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

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

(A) 1

(B) 2

(C) 3

(D) 4