Question 31: Let be a finite set. If is a one-one function, show that is onto also. 

Answer:

In order to prove that f is onto function, we will have to show that every element in A (co-domain) has its pre-image in the domain . In other words, range of

Since is a one-one function. Therefore, are distinct elements of set .

But, has only elements. Therefore, i.e. Co-domain=Range.

Hence, is onto.

Question 32: Let be a finite set. If is an onto function, show that is one-one also.

Answer:

Let

In order to prove that is a one-one function, we will have to show that are distinct elements of

Clearly, Range of

Since, is an onto function. Therefore,

Range of

But is a a finite set consisting of elements. Therefore, are distinct elements of . Hence, is one-one.

Question 33: Let be the set of real numbers. If Then, find   Also show that

Answer:

Clearly; range of is a subset of domain of and range of is a subset of domain of .

 

Answer:

We have, Range of

Clearly, it is a subset of domain of . Hence, exists and such that

Answer:

Clearly,

                                                                                                                [CBSE 2014]

Answer:

Answer:

Question 38: If are defined respectively by , find

Answer:

Answer:

Answer:

Answer:

Answer:

Answer:

We have,

Therefore, for any we have

Answer:

Answer:

We have,

Question 46: Let be a function such that .Show that is onto if and only if is one-one. Describe in this case.

Answer:

Let be onto. Then, we have to prove that is one-one.