Let T: X → Y be a transformation. Recall that T is one-to-one (injective) if any target element can be the image of at most one source element. More specifically, by the usual definition, this means
T(x) = T(x') implies x = x'.
The following characterizes one-to-one transformations. More properties can be found in this exercise.
A transformation T: X → Y is one-to-one ⇔ There is a transformation S: Y → X, such that ST = id_{X}.
Proof T(x) = T(x') implies ST(x) = ST(x'). If ST = id_{X}, then this further impliesx = x'. This proves the ⇐ direction.
Conversely, suppose T is one-to-one. Then for any y ∈ Y, we choose S(y) ∈ X according to to cases
Now starting from x ∈ X, we apply T to get y = T(x). Since T is one-to-one, this x is the unique element with y as the image. Thus we are in the first case above and get ST(x) = S(y) = x. Since this is true for any x, we conclude ST = id_{X}.