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Binary Operations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
'*' on Z defined by a * b = a + b - ab for all a, b ∈ Z.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{Z}.$ Then,
a * b = a + b - ab
= b + a - ba
= 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 - bc)
= a + b + c - bc - a(b + c - bc)
= a + b + c - bc - ab - ac + abc
(a * b) * c = (a + b - ab) * c
= a + b - ab + c - (a + b - ab)c
= a + b + c - ab - ac - bc + abc
Therefore,
a * (b * c) = (a * b) * c, $\forall\ \text{a, b, c}\in\text{Z}$
Thus, * is associative on Z.

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