Check the commutativity and associativity of the following binary… - Vidyadip

Question

Check the commutativity and associativity of the following binary operations:
'*' on N defined by a * b = gcd(a, b) for all a, b ∈ N.

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Answer

Commutativity: Let $\text{a, b}\in\text{N}.$ Then,a * b = gcd(a, b)
= gcd(b, a)
= b * a
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{N}$
Thus '*' is commutative on N.
Associativity: Let $\text{a, b, c}\in\text{N}.$ Then,
a * (b * c) = a * [gcd(a, b)]
= gcd(a, b, c)
(a * b) * c = [gcd(a, b)] * c
= gcd(a, b, c)
Therefore,
a * (b * c) = (a * b) * c, $\forall\ \text{a, b, c}\in\text{N}$
Thus, '*' is not associative on N.

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