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Find identity element
Given:
\( a * b = a + b + 2, \quad a,b \in \mathbb{Z} \)
Identity Definition:
\( a * e = a = e * a \)
Find \( e \):
\( a * e = a + e + 2 = a \)
Simplify:
\( a + e + 2 = a \Rightarrow e = -2 \)
Verification:
\( a * (-2) = a – 2 + 2 = a \)
\( (-2) * a = -2 + a + 2 = a \)
Conclusion:
✔ Identity element = -2