📺 Watch Video Explanation:
Prove that the operation is neither commutative nor associative
Given:
\( a * b = a + 3b – 4, \quad a,b \in \mathbb{Z} \)
1. Not Commutative:
Take \( a = 1 \), \( b = 2 \)
\( a*b = 1 + 3(2) – 4 = 3 \)
\( b*a = 2 + 3(1) – 4 = 1 \)
\( a*b \neq b*a \)
❌ Not commutative
2. Not Associative:
Take \( a = 1 \), \( b = 2 \), \( c = 3 \)
LHS:
\( (a*b)*c = 3 * 3 = 3 + 3(3) – 4 = 8 \)
RHS:
\( a*(b*c) = 1*(2*3) \)
\( 2*3 = 2 + 9 – 4 = 7 \)
\( 1*7 = 1 + 21 – 4 = 18 \)
\( (a*b)*c \neq a*(b*c) \)
❌ Not associative
Conclusion:
❌ The operation is neither commutative nor associative on \( \mathbb{Z} \).