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Question
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Solution
Let e be the identity element in I+ with respect to * such that
\[a * e = a = e * a, \forall a \in I^+ \]
\[a * e = a\text{ and }e * a = a, \forall a \in I^+ \]
\[a + e = a \text{ and }e + a = a, \forall a \in I^+ \]
\[e = 0 , \forall a \in I^+\]
Thus, 0 is the identity element in I+ with respect to *.
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