### Transformation

##### 6. Formal theory of transformation

Using the formal definition, for a transformation **T**: **X** → **Y**, onto means

For any `y` ∈ **Y**, there is `x` ∈ **X**, such that (`x`, `y`) ∈ **T**,

and one-to-one means

(`x`, `y`) and (`x'`, `y`) ∈ **T** implies `x` = `x'`.

The composition of **S**: **X** → **Y** and **T**: **Y** → **Z** is the subset

{(`x`, `z`): There is `y` ∈ **Y** such that (`x`, `y`) ∈ **S** and (`y`, `z`) ∈ **T**}.

A transformation **S**: **Y** → **X** is the inverse of a transformation **T**: **X** → **Y** if the composition `TS` is the diagonal of **Y** and `ST` is the diagonal of **X**.

Technically, if the logical rigor is your only concern, then proofs about transformations should use the formal definition. However, this is not recommended (and should even be discouraged) because such proofs are usually very dry and not intuitive. For an example, see the following proof of (the ⇐ direction of this characterization)

onto and one-to-one ⇒ invertible.

Proof Let **T** ⊂ **X**×**Y** be an onto and one-to-one transformation. Consider the subset

**S** = {(`y`, `x`): (`x`, `y`) ∈ **T**} ⊂ **Y**×**X**.

We verify that the subset satisfies the condition in the formal definition:

For any `y` ∈ **Y**, there is a unique `x` ∈ **X** such that (`y`, `x`) ∈ **S**.

By the construction of **S**, this is the same as

For any `y` ∈ **Y**, there is a unique `x` ∈ **X** such that (`x`, `y`) ∈ **T**.

In the statement above, the existence of `x` means exactly **T** is onto, and the uniqueness of `x` means exactly **T** is one-to-one. Therefore we have

**S** is a transformation ⇔ **T** is onto and one-to-one.

Now starting with any `x` ∈ **X**, since **T** is a transformation, we have (`x`, `y`) ∈ **T** for some (unique) `y` ∈ **Y**. Then by the construction of **S**, we have (`y`, `x`) ∈ **S**. Combining (`x`, `y`) ∈ **T** and (`y`, `x`) ∈ **S**, we deduce that (`x`, `x`) is in the composition subset `ST` ⊂ **X**×**X**. Thus we have proved that the diagonal `Δ`_{X} is a subset of `ST`. Since the diagonal `Δ`_{X} is the identity transformation `id`_{X}: **X** → **X**, by this result we conclude that `ST` = `id`_{X}. By the same reason, we also have `TS` = `id`_{Y}.
Therefore **S** is the inverse of **T**.