UGC NET CSE | October 2022 | Part 1 | Question: 41

Question

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 :

• A. $(\text{A})$ only true
• B. $(\text{B})$ only true
• C. Both $(\text{A})$ and $(\text{B})$ true,
• D. Neither $(\text{A})$ nor $(\text{B})$ true,

Answer 1

(C) 

$a*a*a = a*a*a$

$a*(a*a) = (a*a)*a$    [since semigroup obeys assoc]

$a*b=b*a$

Now, since closure has to be satisfied, either $a*b = a$ or $a*b = b$

1. If $a*b = a$,

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

2. If $a*b = b$,

$b*b = a*a*b = a*b = b$

Credits: @PigeonHole