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Research draft

injection

vr.tr.injection · XCT.QLT

Let an agent explain injective functions, test whether a function is injective, describe injections in algebra and analysis, and separate the medical sense of injection.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain injective functions, test whether a function is injective, describe injections in algebra and analysis, and separate the medical sense of injection.

In mathematics, a function that maps distinct elements of its domain to distinct elements of its codomain, also called a one-to-one or injective function, contrasted with surjections and bijections, with instances such as injective homomorphisms, field and algebra extensions viewed as embeddings, faithful representations and univalent functions in complex analysis; injectivity is a basic property used across algebra, analysis and computing.

What it is for: One-to-one mappings.

It can be explain injectivity; test functions for injectivity; describe injections in algebra and analysis; separate other senses.

Distinguishing features

One-to-one

Left-unique relation

Composable

Embedding role

What it looks like

Not physical; a property of functions.

How it is recognised

Distinct inputs give distinct outputs

Passes the horizontal line test for real functions

A surjection covers the whole codomain

Related models

is a kind of - category

function

is a kind of - category

left-unique relation

is related to - bijections

permutation

is related to - injective homomorphisms

homomorphism

In practice

Families and kinds

injective functions between sets

injective homomorphisms and embeddings

field and algebra extensions as injections

faithful representations

univalent functions

Standards and regulation

Mathematical notation conventions

Failure modes and hazards

Confusing injective with surjective

Confusing with medical or code injection

Errors in proofs of injectivity

Also called

algebra extensioninjective homomorphismradicial morphismunivalent functionfaithful representationfield extensionstrictly monotonic functionOre extensionintegral extensionseparable extensionnormal extensionsimple extensiontranscendental extensionfinitely generated extensionalgebraic extensioninseparable extensionOre algebraGalois extensionseparable closurenormal closurequadratic extensionalgebraic function fieldpurely transcendental extensionalgebraic closureradical extensionabelian extensionGalois closurerational function fieldalgebraic closure of a finite fieldcyclic extension

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Define What an injection is.

Concepts.

Definition

Definition and contrast.

Definition

Definition.

  1. What is an injective function, and how does it differ from surjective and bijective functions? definition
  2. Is the mathematical sense, a medical injection, fuel injection or code injection meant? boundary

Test

Testing injectivity.

Test

Testing.

  1. Is this function injective, and how can it be proved or disproved? action
  2. How does the horizontal line test work? definition
Algebra Injections in algebra.

Structures.

Embeddings

Injective homomorphisms.

Embeddings

Embeddings.

  1. What are injective homomorphisms and embeddings, and how do field and algebra extensions relate? definition
  2. What is a faithful representation? definition

Properties

Properties of injections.

Properties

Properties.

  1. How do injections behave under composition, and what are left inverses? definition
  2. How do injections relate to cardinality? definition
Analysis Injections in analysis and computing.

Applications.

Univalent

Univalent functions.

Univalent

Univalent functions.

  1. What are univalent functions in complex analysis, and why do they matter? definition
  2. Which entry fits complex analysis? action

Computing

Injectivity in computing.

Computing

Computing.

  1. How does injectivity matter for hashing, encodings and data mappings? definition
  2. How can injectivity be checked in code? action
Learn Teaching and history.

Education.

Teach

Teaching injections.

Teach

Teaching.

  1. How can injective, surjective and bijective functions be taught with diagrams? action
  2. Which misconceptions arise? provenance

History

History.

History

History.

  1. How did the terminology of injections and surjections develop? provenance
  2. Which references are standard? provenance

What the second pass must settle

  • Should bijection be a single entry?
  • How should textbooks be linked?
  • How should other senses be split?
Draft as JSON