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Question
Let * be a binary operation on N defined by a * b = a + b + 10 for all a, b ∈ N. The identity element for * in N is _____________ .
Options
−10
0
10
non-existent
Solution
non-existent
Let e be the identity element in N with respect to * such that
\[a * e = a = e * a, \forall a \in N\]
\[a * e = a \text{ and }e * a = a, \forall a \in N\]
\[ \text {Then }, \]
\[a + e + 10 = a \text{ and } e + a + 10 = a, \forall a \in N\]
\[e = - 10 \not\in N\]
So, the identity element with respect to * does not exist in N.
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