Identity Element

An identity or neutral element is an element of a set that, when combined through a binary operation with any element of the same set, leaves the element with which it is combined unchanged

In symbols, given a set 𝑆, in which the binary operation * is possible, and two elements of the set 𝑠 and 𝑒, if it is true that

then 𝑒 is the identity element for the operation * in 𝑆.

Uniqueness of the identity element

A numeric set cannot have two different identity elements for the same binary operation. In fact, if for the operation * in 𝑆 there were two identity elements 𝑒 and 𝑓, then:

since 𝑓 is an identity element. But at the same time,

since 𝑒 is also an identity element. It follows that:

It proves that the identity element must be unique concerning the binary operation * in the set 𝑆.

Additive identity

With regard to addition, the identity element, also known as the additive identity, is 0. This applies to the following numerical sets:

• the set of natural numbers ℕ, if 0 is included in the set;