📺 Watch Video Explanation:
Check commutativity and associativity
Given:
\( a * b = a + b – 7, \quad a,b \in \mathbb{R} \)
Commutativity:
\( a * b = a + b – 7 \)
\( b * a = b + a – 7 = a + b – 7 \)
✔ Operation is commutative
Associativity:
LHS:
\( (a*b)*c = (a + b – 7) * c = (a + b – 7) + c – 7 = a + b + c – 14 \)
RHS:
\( a*(b*c) = a * (b + c – 7) = a + (b + c – 7) – 7 = a + b + c – 14 \)
Thus:
\( (a*b)*c = a*(b*c) \)
✔ Operation is associative
Conclusion:
✔ The operation is both commutative and associative on \( \mathbb{R} \).