# NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 2

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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

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## 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