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Question
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
Solution
Let e be the identity element in Z with respect to * such that
\[a * e = a = e * a, \forall a \in Z\]
\[a * e = a \text{ and }e * a = a, \forall a \in Z\]
\[a + e - 5 = a \text{ and } e + a - 5 = a, \forall a \in Z\]
\[e = 5, \forall a \in Z\]
Thus, 5 is the identity element in Z with respect to *.
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