Difference between revisions of "Identity map"
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If we are dealing with two sets {{M|X}} and {{M|Y}}, then technically we must use differing notation for the identity map on each, for example {{M|\text{Id}_X}} and {{M|\text{Id}_Y}} however this is rarely needed and we (even I, Alec) usually just write {{M|\text{Id} }} for both | If we are dealing with two sets {{M|X}} and {{M|Y}}, then technically we must use differing notation for the identity map on each, for example {{M|\text{Id}_X}} and {{M|\text{Id}_Y}} however this is rarely needed and we (even I, Alec) usually just write {{M|\text{Id} }} for both | ||
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| + | An "identity map" between different sets, for example {{M|f:X\rightarrow Y}} such that {{M|\forall x\in X[f(x)\eq x]}} and as a result we must have {{M|X\subseteq Y}}, then {{M|f}} is called an [[inclusion map]] | ||
==Other notations== | ==Other notations== | ||
Sometimes {{M|I}} is used for the identity map. | Sometimes {{M|I}} is used for the identity map. | ||
| − | + | ==See also== | |
| + | * [[Inclusion map]], which is a map {{M|i:A\rightarrow B}} where {{M|A}}[[subset|{{M|\subseteq}}]]{{M|B}} such that {{M|i:a\mapsto a}} for all {{M|a\in A}} - a sort of identity map in some sense. | ||
==References== | ==References== | ||
<references/> | <references/> | ||
Latest revision as of 15:06, 15 December 2017
Definition
The "identity map", written on this project as [ilmath]\text{Id} [/ilmath], is a map which maps every item (in the domain) to itself, that is if [ilmath]\text{Id}:X\rightarrow X[/ilmath] is a function / map on some set [ilmath]X[/ilmath], then:
- [ilmath]\forall x\in X[\text{Id}(x)\eq x][/ilmath]
Conventions
If we are dealing with two sets [ilmath]X[/ilmath] and [ilmath]Y[/ilmath], then technically we must use differing notation for the identity map on each, for example [ilmath]\text{Id}_X[/ilmath] and [ilmath]\text{Id}_Y[/ilmath] however this is rarely needed and we (even I, Alec) usually just write [ilmath]\text{Id} [/ilmath] for both
An "identity map" between different sets, for example [ilmath]f:X\rightarrow Y[/ilmath] such that [ilmath]\forall x\in X[f(x)\eq x][/ilmath] and as a result we must have [ilmath]X\subseteq Y[/ilmath], then [ilmath]f[/ilmath] is called an inclusion map
Other notations
Sometimes [ilmath]I[/ilmath] is used for the identity map.
See also
- Inclusion map, which is a map [ilmath]i:A\rightarrow B[/ilmath] where [ilmath]A[/ilmath][ilmath]\subseteq[/ilmath][ilmath]B[/ilmath] such that [ilmath]i:a\mapsto a[/ilmath] for all [ilmath]a\in A[/ilmath] - a sort of identity map in some sense.
