Identity map

Definition

The "identity map", written on this project as $\text{Id}$, is a map which maps every item (in the domain) to itself, that is if $\text{Id}:X\rightarrow X$ is a function / map on some set $X$, then:

• $\forall x\in X[\text{Id}(x)\eq x]$

Conventions

If we are dealing with two sets $X$ and $Y$, then technically we must use differing notation for the identity map on each, for example $\text{Id}_X$ and $\text{Id}_Y$ however this is rarely needed and we (even I, Alec) usually just write $\text{Id}$ for both

An "identity map" between different sets, for example $f:X\rightarrow Y$ such that $\forall x\in X[f(x)\eq x]$ and as a result we must have $X\subseteq Y$, then $f$ is called an inclusion map

Other notations

Sometimes $I$ is used for the identity map.

See also

• Inclusion map, which is a map $i:A\rightarrow B$ where $A$$\subseteq$$B$ such that $i:a\mapsto a$ for all $a\in A$ - a sort of identity map in some sense.

References

Grade: D

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Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
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Not really important Alec (talk) 15:04, 15 December 2017 (UTC)