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