Question:
The binary operation \( * \) defined on \( \mathbb{N} \) by:
\[ a * b = a + b + ab \]
Check its properties.
Options:
- (a) Commutative only
- (b) Associative only
- (c) Commutative and associative both
- (d) None of these
Solution:
Step 1: Check Commutativity
\[ a * b = a + b + ab \] \[ b * a = b + a + ba \]
Since addition and multiplication are commutative:
\[ a * b = b * a \]
So, operation is commutative.
—Step 2: Check Associativity
\[ (a * b) * c = (a + b + ab) * c \]
\[ = (a + b + ab) + c + (a + b + ab)c \]
\[ = a + b + c + ab + ac + bc + abc \]
\[ a * (b * c) = a * (b + c + bc) \]
\[ = a + (b + c + bc) + a(b + c + bc) \]
\[ = a + b + c + ab + ac + bc + abc \]
Both sides are equal ⇒ operation is associative.
—Final Answer:
\[ \boxed{\text{(c) Commutative and associative both}} \]