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

Question

Check the commutativity and associativity of the following binary operations:
'*' on Z defined by a * b = a - b for all a, b ∈ Z.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Z}.$ Then,a * b = a - b
b * a = b - a
Therefore,
$\text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Thus '*' is not commutative on Z.
Associativity: Let $\text{a, b, c}\in\text{Z}.$ Then,
a * (b * c) = a * (b - c)
= a - (b - c)
= a - b + c
(a * b) * c = (a - b) - c
= a - b - c
Therefore,
$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})\neq(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}$
Thus, '*' is not associative on Z.

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