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# For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative. On Q, define a ∗ b = ab + 1

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(ii) On $Q$, define $a * b = ab + 1$

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(ii) On $Q$, define $a * b = ab + 1$

ab = ba for all $a,b \in Q$

ab+1 = ba + 1 for all $a,b \in Q$

$\Rightarrow$               $a\ast b=b\ast a$       for $a,b \in Q$

$(1*2)*3 = (1\times 2+1) * 3 = 3 * 3 = 3\times 3+1 = 10$

$1*(2*3) = 1 * (2\times 3+1) = 1 * 7 = 1\times 7+1 = 8$

$\therefore$             $(1\ast 2)\ast 3\neq 1\ast (2\ast 3)$ $;$     where $1,2,3 \in Q$

$\therefore$  operation * is not associative.

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