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Given:
\( a*b = a + b – ab, \quad a,b \in \mathbb{R} \setminus \{1\} \)
Commutativity:
\( a*b = a + b – ab = b + a – ba = b*a \)
✔ Commutative
Associativity:
\( (a*b)*c = a + b + c – ab – bc – ca + abc \)
\( a*(b*c) = a + b + c – ab – bc – ca + abc \)
✔ Associative
Identity Element:
\( a*e = a \Rightarrow a + e – ae = a \Rightarrow e(1-a)=0 \)
Since \( a \neq 1 \), \( (1-a)\neq 0 \), so:
\( e = 0 \)
✔ Identity = 0
Inverse Element:
Find \( b \) such that:
\( a*b = 0 \)
\( a + b – ab = 0 \Rightarrow b(1-a) = -a \)
\( b = \frac{a}{a-1} \)
✔ Inverse of \( a \) = \( \frac{a}{a-1} \)
✔ Every element is invertible
Conclusion:
✔ Commutative & Associative
✔ Identity = 0
✔ Inverse of \( a \) = \( \frac{a}{a-1} \)