Question
Check the commutativity and associativity of the following binary operations:
'*' on R defined by a * b = a + b - 7 for all a, b ∈ R.
'*' on R defined by a * b = a + b - 7 for all a, b ∈ R.
✓
Answer
Commutativity: Let $\text{a, b}\in\text{R}.$ Then,a * b = a + b - 7
= b + a - 7
= b * a
⇒ a * b = b * a
⇒ * is commutative on R.
Associativity: Let $\text{a, b, c}\in\text{R}.$ Then,
(a * b) * c = (a + b - 7) * c
= a + b - 7 + c - 7
= a + b + c - 14 .......(i)
and a * (b * c) = a * (b + c - 7)
= a + b + c - 7 - 7
= a + b + c - 14 ......(ii)
From (i) and (ii)
(a * b) * c = a *( b * c)
* is associative on R.
Thus, * is associative on R.
= b + a - 7
= b * a
⇒ a * b = b * a
⇒ * is commutative on R.
Associativity: Let $\text{a, b, c}\in\text{R}.$ Then,
(a * b) * c = (a + b - 7) * c
= a + b - 7 + c - 7
= a + b + c - 14 .......(i)
and a * (b * c) = a * (b + c - 7)
= a + b + c - 7 - 7
= a + b + c - 14 ......(ii)
From (i) and (ii)
(a * b) * c = a *( b * c)
* is associative on R.
Thus, * is associative on R.
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