One-to-One

Let $f$ be a Function defined on a Set $S$ and taking values in a set $T$. Then $f$ is said to be one-to-one (a.k.a. an Injection or Embedding) if, whenever $f(x)=f(y)$, it must be the case that $x=y$. In other words, $f$ is one-to-one if it Maps distinct objects to distinct objects.

If the function is a linear Operator which assigns a unique Map to each value in a Vector Space, it is called one-to-one. Specifically, given a Vector Space $\Bbb{V}$ with ${\bf X}, {\bf Y}\in \Bbb{V}$, then a Transformation $T$ defined on $\Bbb{V}$ is one-to-one if $T({\bf X}) \not= T({\bf Y})$ for all ${\bf X} \not= {\bf Y}$.

See also Bijection, Onto