📺 Watch Video Explanation:
Given:
\( a * b = a + b – 4, \quad a,b \in \mathbb{Z} \)
i. Commutativity:
\( a*b = a + b – 4 = b + a – 4 = b*a \)
✔ Commutative
Associativity:
LHS:
\( (a*b)*c = (a + b – 4)*c = a + b + c – 8 \)
RHS:
\( a*(b*c) = a*(b + c – 4) = a + b + c – 8 \)
✔ Associative
ii. Identity Element:
\( a * e = a \Rightarrow a + e – 4 = a \Rightarrow e = 4 \)
✔ Identity = 4
iii. Invertible Elements:
Find \( b \) such that:
\( a * b = e = 4 \)
\( a + b – 4 = 4 \Rightarrow a + b = 8 \Rightarrow b = 8 – a \)
✔ Inverse of \( a \) is \( 8 – a \)
✔ Every integer is invertible
Conclusion:
✔ Operation is commutative & associative
✔ Identity = 4
✔ Inverse of \( a \) = \( 8 – a \)