# NCERT Class 12-Mathematics: Chapter – 1 Relations and Functions Part 11 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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)**