Let $(\{a,b\},*)$ be a semigroup, where $a*a=b$.

(A) $a*b=b*a$

(B) $b*b=b$

Choose the most appropriate answer from the options given below:

Given,

$(\{a,b\},*)$ is a semigroup.

So, the operation $*$ is associative.

That means,

$(x*y)*z=x*(y*z)$

Also given,

$a*a=b$

Now check statement (A):

Using associativity,

$(a*a)*a=a*(a*a)$

Since $a*a=b$, we get:

$b*a=a*b$

So,

$a*b=b*a$

Therefore, statement (A) is true.

Now check statement (B):

We need to prove $b*b=b$.

Since $b=a*a$, we can write:

$b*b=(a*a)*b$

Using associativity,

$(a*a)*b=a*(a*b)$

Now, $a*b$ must be either $a$ or $b$ because the set is $\{a,b\}$.

Case 1:

If $a*b=a$, then

$a*(a*b)=a*a=b$

Case 2:

If $a*b=b$, then

$a*(a*b)=a*b=b$

In both cases,

$b*b=b$

Therefore, statement (B) is also true.

Hence, both (A) and (B) are true.