**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

*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

* 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.

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कभी हँसाती है तो कभी रुलाती है ,

पर जो हर हाल में खुश रहते हैं ,

जिंदगी उन्ही के आगे सर झुकाती है।