# NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 4**:

Let the function be defined by . Then, show that is one-one.

**Answer**:

For any two elements , such that , we have

Hence is one-one.

**Question 5**:

If , write .

**Answer**:

**Question 6**:

Let be the function defined by . Then write .

**Answer**:

Given that , then

Hence

**Question 7**:

Is the binary operation defined on (set of integer) by commutative?

**Answer**:

No. Since for , , while so that .

**Question 8**:

If and , write the range of and .

**Answer**:

The range of and the range of .

**Question 9**:

If and , are relations corresponding to the subset of indicated against them, which of , is a function? Why?

**Answer**:

is a function since each element of A in the first place in the ordered pairs is related to only one element of A in the second place while is not a function because is related to more than one element of A, namely, and .

**Question 10**:

If and , show that is one-one from A onto A. Find .

**Answer**:

is one-one since each element of A is assigned to distinct element of the set A. Also, is onto since . Moreover, .

**Question 11**:

In the set of natural numbers, define the binary operation by . Is the operation commutative and associative?

**Answer**:

The operation is clearly commutative since

.

It is also associative because for , we have

.

## Long Answer (L. A.)

**Question 12**:

In the set of natural numbers , define a relation R as follows: if on division by each of the integers and leaves the remainder less than , i.e.. one of the numbers and . Show that is equivalence relation. Also, obtain the pairwise disjoint subsets determined by .

**Answer**:

R is reflexive since for each , . R is symmetric since if , then for , . Also, R is transitive since for , if and , then . Hence is an equivalence relation in which will partition the set into the pairwise disjoint subsets. The equivalent classes are as mentioned below:

It is evident that the above five sets are pairwise disjoint and