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If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________

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If a * b = a such that a ∗ (b ∗ c) = a ∗ b = a and (a * b) * c = a * b = a then ________

(a) * is associative

(b) * is commutative

(c) * is closure

(d) * is abelian

The question was asked by my school teacher while I was bunking the class.

This intriguing question originated from Group Axioms in division Groups of Discrete Mathematics

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Right option is (a) * is associative

Explanation: ‘∗’ can be defined by the formula a∗b = a for any a and b in S. Hence, (a ∗ b)∗c = a∗c = a and a ∗(b ∗ c)= a ∗ b = a. Therefore, ”∗” is associative. Hence (S, ∗) is a semigroup. On the contrary, * is not associative since, for example, (b•c)•c = a•c = c but b•(c•c) = b•a = b Thus (S,•) is not a semigroup.

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