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Question
On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
Solution
To find the identity element, let e be the identity element in Z with respect to * such that
\[a * e = a = e * a, \forall a \in Z\]
\[a * e = a \text{ and }e * a = a, \forall a \in Z\]
\[\text{Then}, \]
\[a + e + 2 = a \text{ and }e + a + 2 = a, \forall a \in Z\]
\[e = - 2 \in Z, \forall a \in Z\]
Thus, -2 is the identity element in Z with respect to *.
Now,
\[\text{ Let }b \in Z \text{ be the inverse of }4 . \]
\[\text{Here},\]
\[4 * b = e = b * 4\]
\[4 * b = e \text{ and }b * 4 = e\]
\[Then, \]
\[4 + b + 2 = - 2 \text{ and }b + 4 + 2 = - 2\]
\[b = - 8 \in Z\]
\[\text {Thus, - 8 \is the inverse of } 4.\]
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