Let * be a binary operation on Z defined by a * b = a + b - 4 for… - Vidyadip

Question

Let * be a binary operation on Z defined by a * b = a + b - 4 for all a, b ∈ Z.
Show that '*' is both commutative and associative.

✓

Answer

Commutativity: Let $\text{a, b}\in\text{Z}.$ Then,
a * b = a + b - 4
= b + a - 4
= b * a
Therefore,
a * b = b * a, $\forall\ \text{a, b}\in\text{Z}$
Thus, * is commutative on Z.
Associativity: Let $\text{a, b, c}\in\text{Z}.$ Then,
a * (b * c) = a * (b + c - 4)
= a + b + c - 4 - 4
= a + b + c - 8
(a * b) * c = (a + b - 4) * c
= a + b - 4 + c - 4
= a + b + c - 8
Therefore, a * (b * c) = (a * b) * c, $\forall\ \text{a, b, c}\in\text{Z}.$
Thus, * is associative on Z.

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