Logics for this lecture:

• Functions $\subseteq$ Relations
• A relation from non-empty set A to non-empty set B is said to be function, if
→ every element of set A has only one image in set B.
In other words, Domain = A and
no distinct ordered pairs have same first element.

DIFFERENT MAPPINGS OF FUNCTIONS

One-One / Injective Mapping / Monomorphism

A function is said to be one-one if each member of the range of the function arises for one and only one member of the domain of the function.
• A function f: A → B is said to be one-one if  $f(a) = f(b) \Rightarrow a = b$

Many-One

A function is said to be many-one if at least one member of the range of the function arises for more than one member of the domain of the function.
• A function f: A → B is said to be one-one if  $f(a) = f(b) \Rightarrow a \ne b$

Onto Mapping / Surjective Function

• In a Surjective function,
Range = Co-domain

Into Mapping

• In a Surjective function,
$Range \subset Co-domain$

• A function which is both one-one and onto is called Bijective Function.

## PART - 1 Play Video

One – One or Injective and Many – One

Question 3. 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.

Onto (Surjective) and Into

## PART - 2 Play Video

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.

जिंदगी बहुत कुछ सिखाती है ,
कभी हँसाती है तो कभी रुलाती है ,
पर जो हर हाल में खुश रहते हैं ,
जिंदगी उन्ही के आगे सर झुकाती है।