The binary operation * defined on N by a^ * b = a + b + ab for al… - Vidyadip

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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