Definition:Identity Arithmetic Function

Definition

The identity arithmetic function $\iota: S \to \Z$ is defined for $n \ge 1$ by:

$\forall n \in S: \map \iota n = \delta_{n 1}$

where:

$S$ is (in theory) any set, but in this context is usually one of the standard number sets $\Z, \Q, \R, \C$

$\delta$ is the Kronecker delta.

That is:

$\forall n \in S: \map \iota n = \begin {cases}

1 & : n = 1\\ 0 & : n \ne 1 \end {cases}$

Also see

The identity arithmetic function can be expressed in terms of the characteristic function $\chi_E: S \to \set {0, 1}$ where $E = \set 1$:

$\forall n \in S: \map \iota n = \map {\chi_{\set 1} } n$

but strictly speaking $\iota$ does not equal $\chi_{\set 1}$ because the codomains are different:

$\Cdm \iota = \Z$

$\Cdm {\chi_{\set 1} } = \set {0, 1}$

Note on Name

The name of this function can be confusing. It is clearly not an identity function as such.

In fact it is a function which returns an answer to the question:

Is $n$ equal to the identity element for multiplication? If so, return $1$, otherwise return $0$.