For given binary operation $$ defined below, determine whether $$ is binary, commutative or associative.
(ii)On Q, define $a * b = a b + 1$

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Solution

On Q, define $a * b = a b + 1$
$a b + 1 \in Q$ , so operation $$ is binary
It is known that
ab =ba for $a, b \in Q$
Therefore, ab +1 =ba +1 for $a, b \in Q$
Therefore, $a b = b $ a for $a, b \in Q$
Therefore, the operation $$ is commutative. It can be observed that
$(1 * 2) * 3 = (1 \times 2 + 1) * 3 = 3 * 3 = 3 \times 3 + 1 = 10 1 * (2 * 3) = 1 * (2 \times 3 + 1) = 1 * 7 = 1 \times 7 + 1 = 8$
Therefore, $(1 * 2) * 3 \neq 1 * (2 * 3)$ , where $1, 2, 3 \in Q$
Therefore, the operation $*$ is not associative.

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