### Transformation

##### 3. One-to-one

Let T: XY 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: XY is one-to-one ⇔ There is a transformation S: YX, such that ST = idX.

Proof T(x) = T(x') implies ST(x) = ST(x'). If ST = idX, then this further impliesx = x'. This proves the ⇐ direction.

Conversely, suppose T is one-to-one. Then for any yY, we choose S(y) ∈ X according to to cases

• If y is in the range of T, then S(y) = the unique x satisfying T(x) = y. The uniqueness of x follows from T being one-to-one.
• If y is not in the range of T, then choose S(y) to be any element of X.

Now starting from xX, 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 = idX.