NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 2 (For CBSE, ICSE, IAS, NET, NRA 2022)
Get unlimited access to the best preparation resource for CBSE/Class-12 : get questions, notes, tests, video lectures and more- for all subjects of CBSE/Class-12.
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