For each binary operation * defined below, determine whether * is… - Vidyadip

Question

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

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Answer

For commutativity: a * b = ab + 1 and b * a = ba + 1 = ab + 1 = a * b
For associativity: a * (b * c) = a * (bc + 1) = a(bc + 1) + 1 = abc + a + 1
Also, (a * b) * c = (ab + 1)c + 1 = abc + c + 1
$\therefore\ \ \text{a }*(\text{b }*\text{c})\neq(\text{a }*\text{b })*\text{c}$
Therefore, the operation * is commutative but not associative.

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