A binary operation on a set has always the identity element.

Question

State True or False for the statement:

A binary operation on a set has always the identity element.

Answer 1

False

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.