Definition:Identity Mapping

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Definition

The identity mapping of a set $S$ is the self-map $I_S: S \to S$ defined as:

$I_S = \set {\tuple {x, y} \in S \times S: x = y}$

or alternatively:

$I_S = \set {\tuple {x, x}: x \in S}$


That is:

$I_S: S \to S: \forall x \in S: \map {I_S} x = x$


That is, it is a mapping in which every element is a fixed element.


Also known as

The identity mapping can also be seen referred to as the identity operator, identity function or identity transformation.


Alternative symbols for $I_S$ include $1_S$, $i_S$, $j_S$, $id_S$, $\operatorname {id}_S$, $\operatorname {Id}_S$, $\iota_S$ and $\varepsilon_S$.

The subscript is frequently removed if there is no danger of confusion as to the domain under discussion.


Some sources use the same symbol for the identity mapping as for the inclusion mapping without confusion, on the grounds that the domain and codomain of the latter are different.


As the identity mapping is (technically) exactly the same thing as the diagonal relation, the symbol $\Delta_S$ is often used for both.


Also see

  • Results about identity mappings can be found here.


Sources

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