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

Question

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}.$ Then,a * b = ab + 1
= ba + 1
= b * a
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{Q}$
Thus, * is commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q}.$ Then,
a * (b * c) = a * (bc + 1) = a(bc + 1) + 1
= abc + a + 1
(a * b) * c = (ab + 1) * c
= (ab + 1)c + 1
= abc + c + 1
Therefore,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})\neq(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}$
Thus, * is not associative on Q.

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