math111_logo Transformation

6. Formal theory of transformation

Using the formal definition, for a transformation T: XY, onto means

For any yY, there is xX, such that (x, y) ∈ T,

and one-to-one means

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

The composition of S: XY and T: YZ is the subset

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

A transformation S: YX is the inverse of a transformation T: XY 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 TX×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 yY, there is a unique xX such that (y, x) ∈ S.

By the construction of S, this is the same as

For any yY, there is a unique xX 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 xX, since T is a transformation, we have (x, y) ∈ T for some (unique) yY. 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 STX×X. Thus we have proved that the diagonal ΔX is a subset of ST. Since the diagonal ΔX is the identity transformation idX: XX, by this result we conclude that ST = idX. By the same reason, we also have TS = idY. Therefore S is the inverse of T.

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