# Exercise 3.1

Question 1

Give an example of a function
(i) which is one-one but not onto

Sol :

Injectivity :

Let x and be any two elements in the domain (Z) , such that

So , is one-one

Surjectivity :

Let be any element in the co-domain (Z) , such that for some element in Z (domain) .

it may not be in the domain (Z) because if we take ,

So, for every element in the co domain there need not be any element in the domain such that

Thus , is not onto .

(ii) which is not one-one but onto

Injectivity :

Let x and be any two elements in the domain (Z) , such that

So ,different element of domain may give the same image .

So , is not one-one .

Surjectivity :

Let be any element in the co-domain (Z) , such that for some element in Z (domain) .

, which is an element in Z (domain) .

So, for every element in the co domain there exist a pre-image in the domain .

Thus , is onto .

(iii) which is neither one-one nor onto

Injectivity :

Let x and be any two elements in the domain (Z) , such that

So , different elements of domain may give the same image .

So , is not one-one

Surjectivity :

Let be any element in the co-domain (Z) , such that for some element in Z (domain) .

always .

For example if we take ,

So , may not be in Z (domain)

Thus , is not onto .

Question 2

Which of the following functions from to are one-one and onto ?

(i) ; ;

Sol :

Injectivity :

Every element of has different images in .

So , is one-one .

Surjectivity :

Co-domain of

Range of = set of images

co-domain = range

So , is onto .

(ii) ; ;

Sol :

Injectivity :

Every element of has different images in .

So , is one-one .

Surjectivity :

Co-domain of

Range of = set of images

co-domain = range

So , is onto .

(iii) ; ;

Sol :

Injectivity :

and have same image . Also , and have same image z .

So , is not one-one .

Surjectivity :

Co-domain of

Range of = set of images

So , the co-domain is not equals to range .

So , is  not onto .

Question 3

Prove that the function , is one-one but not onto .

Sol :

Injectivity :

Let and be any two elements in the domain (N) , such that

taking common (x – y)

[(x+y+1) can not be zero because x and y are natural numbers]

So , is one-one

Surjectivity :

The minimum number in is 1 .

When = 1 ,

, for every x in N

will not assume the values 1 and 2

So , is  not onto

Question 4

Let and . Show that is neither one-one nor onto .

Sol :

Given , , and

Injectivity :

1 and -1 have same images .

So , is not one-one

Surjectivity :

Co-domain of

Range of

So , co-domain is not same as range

Hence , is not onto

Question 5

Classify the following functions as injection, surjection or bijection :

(i)

Sol :

Injection :

Let x and y be any elements in the doamin (N) such that

we do not get because x and y are natural numbers

So , is an injection

Surjection :

Let y be any element in co-domain (N) such that for some element x in (N) domain

, which may not be in N

For example , if ,

is not in N

So , is not a surjection

So , is not a bijection