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

Question

The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is:

Commutative only.
Associative only.
Commutative and associative both.
None of these.

✓

Answer

Commutative and associative both.

Solution:

a * b = a + b + ab

b * a = b + a + ba

⇒ 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

⇒ (a * b) * c = a * (b * c)

So * is associative.

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