Definition:Identity Mapping

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The identity mapping of a set $S$ is the mapping $I_S: S \to S$ 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:

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

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

Also known as

The identity mapping is also referred to by some sources 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 which set is 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.