# 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*

Booklets/Notes/Assignments are typed here on this website and their PDFs will be made available soon.

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.