Injection
From Maths
An injective function is 1:1, but not nessasarally onto.
Contents
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:
- [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)=f(x_2)][/math] (the contrapositive of the above)
Notes
The cardinality of the inverse of an element [math]y\in Y[/math] may be no more than 1; that is 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)
TODO: Find reference - should be easy!
