Question

* : R×R - R is given by a*b = 3a^2 - b. Find the value of 2*3.
Is * commutative? Justify your answer.



​

Answer 1

Answer:

The binary operation * ℝ x ℝ → ℝ is given by

a*b = 3a² - b

1. Value of 2*3

2*3 = 3(2)² - 3

= 12 - 3

= 9

2. Proof by Counter Example

Let us take the above example.

a*b = 9

b*a = 3(3)² - 2

= 27 - 2

= 25

a*b and b*a are not equal when we take a = 3 and b = 2. It is therefore enough to conclude that * is not commutative.

Answer 2

Answer :

★ 2*3 = 9

★ * is not commutative

Solution :

Given Operation :

* : R × R → R , a*b = 3a²b - b

★ Value of 2*3 :-

=> 2*3 = 3×2² - 3

=> 2*3 = 3×4 - 3

=> 2*3 = 12 - 3

=> 2*3 = 9

Hence , 2*3 = 9

★Whether * is commutative :-

Note : A binary Operation * is said to be commutative if a*b = b*a .

Now ,

=> LHS = a*b

=> LHS = 3a² - b

Also ,

=> RHL = b*a

=> RHS = 3b² - a

LHS ≠ RHS

Since , a*b ≠ b*a thus the given binary Operation * is not commutative .