# Injection

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This needs to be modified (in tandem with Surjection) to:
1. allow surjection/injection/bijection to be seen through the lens of Category Theory. Alec (talk) 21:50, 8 May 2018 (UTC)
2. be linked to cardinality of sets and that Cantor theorem. Alec (talk) 21:50, 8 May 2018 (UTC)

An injective function is 1:1, but not nessasarally onto.

## Definition

For a function $f:X\rightarrow Y$ every element of $X$ is mapped to an element of $Y$ and no two distinct things in $X$ are mapped to the same thing in $Y$. That is[1]:

• $\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2]$

Or equivalently:

• $\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)\ne f(x_2)]$ (the contrapositive of the above)

Sometimes an injection is denoted [ilmath]\rightarrowtail[/ilmath][2] (and a surjection [ilmath]\twoheadrightarrow[/ilmath] and a bijection is both of these combined (as if super-imposed on top of each other) - there is no LaTeX arrow for this however) - we do not use this convention.

## Notes

### Terminology

• An injective function is sometimes called an embedding[1]
• Just as surjections are called 'onto' an injection may be called 'into'[3] however this is rare and something I frown upon.
• This is French, from "throwing into" referring to the domain, not elements themselves (as any function takes an element into the codomain, it need not be one-to-one)
• I do not like using the word into but do like onto - I say:
"But [ilmath]f[/ilmath] maps [ilmath]A[/ilmath] onto [ilmath]B[/ilmath] so...."
"But [ilmath]f[/ilmath] is an injection so...."
"As [ilmath]f[/ilmath] is a bijection..."
• I see into used rarely to mean injection, and in fact any function [ilmath]f:X\rightarrow Y[/ilmath] being read as [ilmath]f[/ilmath] takes [ilmath]X[/ilmath] into [ilmath]Y[/ilmath] without meaning injection[1][4]

### Properties

• The cardinality of the inverse of an element $y\in Y$ may be no more than 1
• Note this means it may be zero
In contrast to a bijection where the cardinality is always 1 (and thus we take the singleton set $f^{-1}(y)=\{x\}$ as the value it contains, writing [ilmath]f^{-1}(y)=x[/ilmath])

## References

1. Analysis: Part 1 - Elements - Krzysztof Maurin
2. Notes On Set Theory - Second Edition - Yiannis Moschovakis
3. http://mathforum.org/library/drmath/view/52454.html
4. Real and Abstract Analysis - Edwin Hewitt and Karl Stromberg