Question

The binary operation * defined on N by

a * b = a + b + ab for all a, b ∈ N is

(a) commutative only
(b) associative only
(c) commutative and associative both
(d) none of these

Answer 1

(c) commutative and associative both

Commutativity:

$$\begin{gathered} \text{Let}a,b\in N \\ \mathrm{Then}, \\ a*b=a+b+ab \\ =b+a+ba \\ =b*a \\ \text{Thus,} \\ a*b=b*a,\forall a,b\in N \end{gathered}$$

Thus, * is commutative on N.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in N \\ a*\left(b*c\right)=a*\left(b+c+bc\right) \\ =a+b+c+bc+a\left(b+c+bc\right) \\ =a+b+c+bc+ab+ac+abc \\ \left(a*b\right)*c=\left(a+b+ab\right)*c \\ =a+b+ab+c+\left(a+b+ab\right)c \\ =a+b+c+ab+ac+bc+abc \\ \text{Therefore,} \\ a*\left(b*c\right)=\left(a*b\right)*c,\forall a,b,c\in N \end{gathered}$$

Thus, * is associative on N.