📺 Watch Video Explanation:
Check commutativity and associativity
Given:
\( a * b = ab^2, \quad a,b \in \mathbb{Q} \)
Commutativity:
\( a * b = ab^2 \)
\( b * a = ba^2 \)
Clearly:
\( ab^2 \neq ba^2 \quad (\text{in general}) \)
❌ Operation is NOT commutative
Associativity:
LHS:
\( (a*b)*c = (ab^2)*c = (ab^2)c^2 = abc^2 b^2 = a b^2 c^2 \)
RHS:
\( a*(b*c) = a*(bc^2) = a(bc^2)^2 = a b^2 c^4 \)
Since:
\( a b^2 c^2 \neq a b^2 c^4 \)
❌ Operation is NOT associative
Conclusion:
❌ Neither commutative nor associative