We know that, ‘e’ is called an identity element for the binary operation ‘*’ on set S, such that
a * e = e * a = a ∀ a ∈ S
Let ‘+’ be a binary operation on set N.
Now, we need to find e ∈ N such that
n + e = e + n = n ∀ n ∈ N
n + 0 = 0 + n = n ∀ n ∈ N
We observe that addition of any natural number with 0 gives the desired result but 0 ∉ N, hence 0 is not the identity element for the addition operation on N.
∴ The binary operation ‘+’ on N does not have any identity element.
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