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===Commentary Note 11.12===

===Commentary Note 11.12===

Since functions are special cases of 2-adic relations, and since the space of 2-adic relations is closed under relational composition, in other words, the composition of a couple of 2-adic relations is again a 2-adic relation, we know that the relational composition of a couple of functions has to be a 2-adic relation.  If it is also necessarily a function, then we would be justified in speaking of "functional composition", and also of saying that the space of functions is closed under this functional form of composition.

Since functions are special cases of 2-adic relations, and since the space of 2-adic relations is closed under relational composition — in other words, the composition of a pair of 2-adic relations is again a 2-adic relation — we know that the relational composition of two functions has to be a 2-adic relation.  If it is also necessarily a function, then we would be justified in speaking of ''functional composition', and also in saying that the space of functions is closed under this functional form of composition.

Just for novelty's sake, let's try to prove this for relations that are functional on correlates.

Just for novelty's sake, let's try to prove this for relations that are functional on correlates.

So our task is this:  Given a couple of 2-adic relations, ''P'' ⊆ ''X'' × ''Y'' and ''Q'' ⊆ ''Y'' × ''Z'', that are functional on correlates,

The task is this — We are given a pair of 2-adic relations:

''P'' : ''X'' ← ''Y'' and ''Q'' : ''Y'' ← ''Z'', we need to determine whether the relational composition ''P'' o ''Q'' ⊆ ''X'' × ''Z'' is also ''P'' o ''Q'' : ''X'' ← ''Z'', or not.

 

| <math>P \subseteq X \times Y \quad \text{and} \quad Q \subseteq Y \times Z</math>

 

<math>P\!</math> and <math>Q\!</math> are assumed to be functional on correlates, a premiss that we express as follows:

 

| <math>P : X \leftarrow Y \quad \text{and} \quad Q : Y \leftarrow Z</math>

 

We are charged with deciding whether the relational composition <math>P \circ Q \subseteq X \times Z</math> is also functional on correlates, that is, whether <math>P \circ Q : X \leftarrow Z</math> or not.

It always helps to begin by recalling the pertinent definitions.

It always helps to begin by recalling the pertinent definitions.