NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 2 (For CBSE, ICSE, IAS, NET, NRA 2023)

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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