Let ({π,π},β) be a semigroup, where πβπ=π. (A) πβπ=πβπ (B) πβπ=πβ¦
2022
LetΒ ({π,π},β)Β be a semigroup, whereΒ πβπ=π.
(A)Β πβπ=πβπ
(B)Β πβπ=π
Choose the most appropriate answer from the options given below :
- A.
(A)Β only true
- B.
(B)Β only true
- C.
BothΒ (A)Β andΒ (B)Β true
- D.
NeitherΒ (A)Β norΒ (B)Β true
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Correct answer: C
Given: a semigroup with elements a and b and the relation a*a = b.
Apply associativity to (a*a)*a = a*(a*a). Since a*a = b, this becomes b*a = a*b. Therefore a*b = b*a, so the statement a*b = b*a is true.
Let p = a*b (so b*a = p as well). Use associativity on (a*a)*b = a*(a*b). The left side is b*b and the right side is a*p.
If p = a then a*p = a*a = b. If p = b then a*p = a*b = p = b. In either case a*p = b, so b*b = b. Therefore the statement b*b = b is true.
Conclusion: Both a*b = b*a and b*b = b are true.
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