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Binary Operations — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsBinary Operations1 Mark
MCQ
The binary operation $*$ defined on $N$ by $a * b = a + b + ab$ for all $a, b \in N$ is:
  • A
    Commutative only.
  • B
    Associative only.
  • ✓
    Commutative and associative both.
  • D
    None of these.
✓

Answer

Correct option: C.
Commutative and associative both.
$a * b = a + b + ab$
$b * a = b + a + ba$
$\Rightarrow a * b = b * a$
So $*$ is commutative.
Now,
$(a * b) * c$
$= (a + b + ab) * c$
$= a + b + ab + c + ca + cb + abc$
$a * (b * c)$
$= a * (b + c + bc)$
$= a + b + c + bc + ab + ac + abc$
$\Rightarrow (a * b) * c = a * (b * c)$
So $*$ is associative.

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