Question

Find which of the operations given above has identity.

Answer 1

question is ---->
(i) a * b = a − b 
(ii) a * b = a^2 + b^2 
(iii) a * b = a + ab 
(iv) a * b = (a − b)^2 
(v) a * b = ab / 4
(vi) a * b = ab^2 
Find which of the operations given above has identity ?

solution :-
concept :- 1.) An element e ϵ Q will be the identity element for the operation * if
a * e = a = e * a, $\forall$  a ϵ Q.
2.) if operation is not commutative then it doesn't have identity element.

Let's check all operations

(i) a * b = a - b
we can see 1 * 2 = 1 - 2 = -1
2 * 1 = 2 - 1 = 1
1 * 2 ≠ 2 * 1 , where 1, 2 ϵ Q
this is not commutative. therefore , it doesn't have identity element.

(ii) a * b = a² + b²
a * e = a , a² + e² = a
but if a = -2 ϵ Q
e * -2 = e² + (-2)² ≠ -2
therefore , it doesn't have identity element.

(iii) a * b = a + ab
we can see that 1 * 2 = 1 + 1 × 2 = 3
2 * 1 = 2 + 2 × 1 = 4
e.g., 1 * 2 ≠ 2 * 1
therefore , * is not commutative .
so, it doesn't have identity element.

(iv) a * b = (a - b)²
a * e = a , (a - e)² = a
if we take a = 0 , (0 - e)² ≠ 0
so, it doesn't have identity element.

(v) a * b = ab/4
a * e = a , ae/4 = a
therefore , e = 4 is the identity element.
because , a * 4 = 4 * a = 4a/4 = a .

(vi) a * b = ab²
we can see 1 * 2 = 1 × 2² = 4
2 * 1 = 2 × 1² = 2
e.g., 1 * 2 ≠ 2 * 1 , therefore * is not commutative.
hence, it doesn't have identity element.