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Find identity element
Given:
\( a * b = a + b + ab, \quad a,b \in \mathbb{Q} \setminus \{-1\} \)
Identity Definition:
\( a * e = a = e * a \)
Find \( e \):
\( a * e = a + e + ae = a \)
Simplify:
\( a + e + ae = a \Rightarrow e(1 + a) = 0 \)
Since \( a \neq -1 \), we have \( (1+a) \neq 0 \), so:
\( e = 0 \)
Verification:
\( a * 0 = a + 0 + 0 = a \)
\( 0 * a = 0 + a + 0 = a \)
Conclusion:
✔ Identity element = 0