# Â  Â  Â  Â  Â  Â  Â  Exercise 1.2

Question 1: Show that the function defined by is one-one and onto, where the set of all non-zero real numbers. Is the result true, if the domain is replaced by with codomain being same as (Class 12 ncert solution math exercise 1.2 )

Solution:

For one-one:

is one-one.

For onto:

For , there exists as such that

is onto.

Given function is one-one and onto. Consider function â€¢ defined by

We have,

is one-one.

is not onto as for . there exist any in such that

Function is one-one but not onto.

Question 2:Check the injectivity and surjectivity of the following functions:

i. given by

ii. given by

iii.Â  given by

Â ivÂ  given by

v. given by

Solution:i. For given by

is injective. . But, there does not exist any in such that

is not surjective

Function is injective but not surjective.

ii. given by

but

is not injective.

But, there does not exist any such that

is not surjective.

Function is neither injective nor surjective.

iii. given by

but

is not injective.

But, there does not exist any such that

is not surjective.

Function is neither injective nor surjective.

iv. given by

is injective.

. But, there does not exist any in such that

is not surjective

Function is injective but not surjective.

v. given by

is injective.

. But, there does not exist any in such that

is not surjective.

Function is injective but not surjective.

Question 3:Prove that the greatest integer function given by onto, where denotes the greatest integer less than or equal to .

Solution: given by

, but

is not one-one.

Consider

such that

is not onto.

The greatest integer function is neither one-one nor onto.

Question 4:Show that the modulus function given by is neither one-one nor onto, where is , if is positive or 0 and is , if is negative.

Solution: is

and

but

is not one-one.

Consider

is non-negative. There exist any element in domain such that

is not onto.

The modulus function is neither one-one nor onto.

Question 5:Show that the signum function given by onto.

Solution:

, but

is not one-one.

takes only 3 values for the element in co-domain

, there does not exist any in domain such that .

is not onto.

The signum function is neither one-one nor onto.

Question 6: Let and let be a function from to B. Show that is one-one.

Solution:

is defined as

It is seen that the images of distinct elements of under are distinct.

is one-one.

Question 7: In each of the following cases, state whether the function is one-one, onto or bijective.

Justify your answer.

i). defined by

ii). defined by

Solution: i. defined by

such that

is one-one.

For any real number in , there exists in such that

is onto.

Hence, is bijective.

ii. defined by

such that

does not imply that

Consider

is not one-one.

Consider an element in co domain .

It is seen that is positive for all .

is not onto.

Hence, is neither one-one nor onto.

Question 8: Let and be sets. Show that such that is a bijective function.

Solution: is defined as .

and

is one-one.

there exist such that

is onto.

is bijective.

Question 9: Let be defined as

for all .

State whether the function is bijective. Justify your answer.

Solution: be defined as

and

, where

is not one-one.

Consider a natural number in co domain .

Case I: is odd

for some there exists such that

Case II: is even

for some there exists such that

is onto.

is not a bijective function.

Question 10: Let and defined by . Is one-one and onto? Justify your answer.

Solution:

is one-one.

Let , then

The function is onto if there exists such that .

Now,

Thus, for any , there exists such that

is onto.

Hence, the function is one-one and onto.

Question 11: Let defined as .Choose the correct answer.
A. is one-one onto
B. is many-one onto
C. is one-one but not onto
D. is neither one-one nor onto

Solution: defined as

does not imply that .

For example

is not one-one.

Consider an element 2 in co domain there does not exist any in domain such that .

is not onto.

Function is neither one-one nor onto.

The correct answer is D.

Question 12: Let defined as .Choose the correct answer.
A. is one-one onto
B. is many-one onto
C. is one-one but not onto
D. is neither one-one nor onto

Solution: defined as

is one-one.

For any real number in co domain , there exist in such that

is onto.

Hence, function is one-one and onto.

The correct answer is A.