NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 11

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Question 26:

Let be the binary operation defined on Q. Find which of the following binary operations are commutative

(i)

(ii)

(iii)

(iv)

Answer:

Given that, be the binary operation defined on Q.

A binary operation is commutative if

(i)

Hence, is not commutative on Q.

(ii)

Hence, is commutative on Q.

(iii)

Hence, is not commutative on Q.

(iv)

Hence, is commutative on .

Question 27:

Let be binary operation defined on R by . Then the operation is

(i) Commutative but not associative

(ii) Associative but not commutative

(iii) Neither commutative nor associative

(iv) Both commutative and associative

Answer:

(i)

Given that,

be binary operation defined on R by

A binary operation is commutative if

Now,

So, is commutative on R.

A binary operation is associative if

Now,

So,is not associative on R.

Hence, is commutative but not associative on R.

Objective Type Questions

Choose the Correct Answer Out of the Given Four Options in Each of the Exercises from 28 to 47 (M.C.Q.)

Question 28:

Let be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as if is congruent to , . Then R is

(A) Reflexive but not transitive

(B) Transitive but not symmetric

(C) Equivalence

(D) None of these

Answer: (C)