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 | ||
| + | |||
| + | 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 Id, is a map which maps every item (in the domain) to itself, that is if Id:X→X is a function / map on some set X, then:
- ∀x∈X[Id(x)=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 IdX and IdY however this is rarely needed and we (even I, Alec) usually just write Id for both
An "identity map" between different sets, for example f:X→Y such that ∀x∈X[f(x)=x] and as a result we must have X⊆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→B where A⊆B such that i:a↦a for all a∈A - a sort of identity map in some sense.
References
Grade: D
This page requires references, it is on a to-do list for being expanded with them.
