NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 7 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 12:

Let and . Find whether the following subsets of are functions from X to Y or not.

(i)

(ii)

(iii)

(iv)

Answer:

(i) is not function

(ii) is function

(iii) is function

(iv) is not function

Question 13:

If functions and satisfy , then show that f is one-one and is onto.

Answer:

Given:

In order to prove that f is one-one, it is sufficient to prove that

Let such that . Then,

Thus, f is one-one.

Now, in order to prove that is onto, it is sufficient to prove that each element in A has pre-image in B.

Let .

Also, is a function

Now,

Let

Thus, for every there exists such that .

is onto.

Question 14:

Let be the function defined by . Then, find the range of .

Answer:

Question 15:

Let be a fixed positive integer. Define a relation R in Z as follows: if and only if is divisible by . Show that R is an equivalence relation.

Answer:

In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.

Given that, , if and only if is divisible by n.

Now,

is divisible by .

is divisible by

Thus, R is reflexive on Z.

R is Symmetric if

is divisible by n

for some

is divisible by

Thus, R is symmetric on Z.

R is Transitive if

is divisible by

for some

is divisible by

for some

Now,

and

where

is divisible by n.

Thus, R is transitive on Z.

Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.