Part - 2 Lecture - 3 Chapter 1 Relations and Functions

“I can’t change the direction of the wind, but I can adjust my sails to always reach my destination.” —Jimmy Dean

Brief of all rules about injectivity and surjectivity

Question 1. Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subsets of X ×Y are functions from X to Y or not.
a. f = {(1, 4), (1, 5), (2, 4), (3, 5)}
b. h = {(1,4), (2, 5), (3, 5)}
c. g = {(1, 4), (2, 4), (3, 4)}
d. k = {(1,4), (2, 5)}.

Question 2. If f : A → B is bijective function such that n(A) =10, then n(B) = ?

Question 4. Let A be the set of all 50 students of Class X in a school. Let f : A → N be function defined by f(x) = roll number of the student x. Show that f is one-one but not onto.

Question 5. 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* ?

Question 6. Check the injectivity and surjectivity of the following functions:
a. f :N→N given by $f(x)=x^2$
b. f : Z→Z given by $f(x)=x^2$
c. f : R→R given by $f(x)=x^2$
d. f : N→N given by $f(x)=x^3$
e. f : Z→Z given by $f(x)=x^3$

Question 7. 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.

Question 8. 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.

Question 9. Show that the Signum Function f: R→R, is neither one-one nor onto.

Question 10. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
a. f : R→R defined by f(x) = 3 – 4x
b. f : R→R defined by $f(x) = 1 + x^2$

Question 11. Let f : R→R be defined as $f(x) = x^4$ . Choose the correct answer.
a. f is one-one onto
b. f is one-one but not onto
c. f is many-one onto
d. f is neither one-one nor onto.

Question 12. Let f : R→R be defined as f(x) = 3x. Choose the correct answer.
a. f is one-one onto
b. f is one-one but not onto
c. f is many-one onto
d. f is neither one-one nor onto.