NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 8

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Long Answer (L. A)

Question 16:

If , define relations on A which have properties of being:

(a) Reflexive, transitive but not symmetric

(b) Symmetric but neither reflexive nor transitive

(c) Reflexive, symmetric and transitive.

Answer:

Given that,

(a) Reflexive, transitive but not symmetric

Let R be a relation defined by

on .

R is reflexive

R is transitive and

R is not symmetric but

Hence, R is reflexive, transitive but not symmetric.

(b) Symmetric but neither reflexive nor transitive

Let R be a relation defined by

on set A.

R is not reflexive

R is symmetric and

R is not transitive and

Hence, R is symmetric but neither reflexive nor transitive.

(c) Reflexive, symmetric and transitive.

Let R be a relation defined by

on set A.

R is reflexive

R is symmetric

R is transitive and

Hence, R is reflexive, symmetric and transitive.

Question 17:

Let R be relation defined on the set of natural number N as follows:

. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive.

Answer:

Domain of and

Range of . R is neither reflective, nor symmetric and nor transitive.

Question 18:

Given . Construct an example of each of the following:

(a) An injective mapping from A to B

(b) A mapping from A to B which is not injective

(c) A mapping from B to A.

Answer:

Given that,

(a) An injective mapping from A to B

Let denote a mapping

Now,

When we get

Similarly, and will give and respectively.

We observe that each element of A has unique image in B.

Thus, f is injective.

(b) A mapping from A to B which is not injective

Let g: denote a mapping such that

We observe that 2 and does not have unique image.

Thus, g is not injective.

(c) a mapping from B to A.

Let denote a mapping such that

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