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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
MCQ
Let $*$ be a binary operation on $R$ defined by $a * b = ab + 1.$ Then, $*$ is:
  • ✓
    Commutative but not associative.
  • B
    Associative but not commutative.
  • C
    Neither commutative nor associative.
  • D
    Both commutative and associative.
✓

Answer

Correct option: A.
Commutative but not associative.
Commutativity:
Let $\text{a, b}\in\text{R}$
$a * b = ab + 1$
$= ba + 1$
$= b * a$
Therefore,
$a * b = b * a, \forall\text{ a, b}\in\text{R}$
Therefore, $*$ is commutative on $R.$
Associativity:
Let $\text{ a, b, c}\in\text{R}$
$a * (b * c) = a * (bc + 1)$
$= a(bc + 1) + 1$
$= abc + a + 1$
$(a * b) * c = (ab + 1) * c$
$= (ab + 1)c + 1$
$= abc + c + 1$
$\therefore a * (b * c) \neq (a * b) * c$
For example: $a = 1, b = 2$ and $c = 3 [$which belong to $R]$
Now,
$1 * (2 * 3) = 1 * (6 + 1)$
$= 1 * 7$
$= 7 + 1$
$= 8$
$(1 * 2) * 3 = (2 + 1) * 3$
$= 3 * 3$
$= 9 + 1$
$= 10$
$\Rightarrow 1 * (2 * 3) \neq (1 * 2) * 3$
Therefore, $\exists a = 1, b = 2$ and $c = 3$ which belong to $R$ such that
$a * (b * c) \neq (a * b) * c$
Hence, $*$ is not associative on $R.$

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