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Question
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?
Solution
Let a ∈ Z and b ∈ Z be the inverse of a. Then,
\[a * b = e = b * a\]
\[a * b = e \text{ and }b * a = e\]
\[a + b - 4 = 4 \text{ and } b + a - 4 = 4\]
\[b = 8 - a \in Z\]
\[\text{Thus},8 - \text{a is the inverse of a} \in Z . \]
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