Question
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = a - b for all a, b ∈ Q.
'*' on Q defined by a * b = a - b for all a, b ∈ Q.
✓
Answer
Commutativity: Let $\text{a, b}\in\text{Q.}$ Then,
a * b = a - b
b * a = b - a
Therefore,
$\text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Thus, * is not commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q.}$ Then,
a * (b * c) = a * (b - c)
= a - (b - c)
= a - b + c
(a * c) * 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 Q.
a * b = a - b
b * a = b - a
Therefore,
$\text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$
Thus, * is not commutative on Q.
Associativity: Let $\text{a, b, c}\in\text{Q.}$ Then,
a * (b * c) = a * (b - c)
= a - (b - c)
= a - b + c
(a * c) * 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 Q.
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