# 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:

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

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] (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.