Injection
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Grade: A*
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This needs to be modified (in tandem with Surjection) to:
 allow surjection/injection/bijection to be seen through the lens of Category Theory. Alec (talk) 21:50, 8 May 2018 (UTC)
 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 [math]f:X\rightarrow Y[/math] every element of [math]X[/math] is mapped to an element of [math]Y[/math] and no two distinct things in [math]X[/math] are mapped to the same thing in [math]Y[/math]. That is^{[1]}:
 [math]\forall x_1,x_2\in X[f(x_1)=f(x_2)\implies x_1=x_2][/math]
Or equivalently:
 [math]\forall x_1,x_2\in X[x_1\ne x_2\implies f(x_1)\ne f(x_2)][/math] (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 superimposed on top of each other)  there is no LaTeX arrow for this however)  we do not use this convention.
Statements
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 onetoone)
 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 [math]y\in Y[/math] 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 [math]f^{1}(y)=\{x\}[/math] as the value it contains, writing [ilmath]f^{1}(y)=x[/ilmath])
 Note this means it may be zero
See also
References
 ↑ ^{1.0} ^{1.1} ^{1.2} Analysis: Part 1  Elements  Krzysztof Maurin
 ↑ Notes On Set Theory  Second Edition  Yiannis Moschovakis
 ↑ http://mathforum.org/library/drmath/view/52454.html
 ↑ Real and Abstract Analysis  Edwin Hewitt and Karl Stromberg
