Combinaision of two functions

Question

Let us denote $X_0 = \{x, y\}$ and $X_1 = \{a, b\}$ two disjoint sets of variables; let us denote $V$ a set of values.

I have two functions $f_0 : X_0 \rightarrow V$ and $f_1 : X_1 \rightarrow V$, for instance: $f_0(x) =1, f_0(y) = 2$ and $f_1(a) =3, f_1(b) = 4$.

The combinaision of $f_0$ and $f_1$ forms another function $f : (X_0 \cup X_1) \rightarrow V$ (e.g., $f(x) =1, f(y) = 2, f(a) =3, f(b) = 4$).

I am looking for an elegant way to express the relation between $f$, $f_0$ and $f_1$.

Some may suggest $f = f_0 \wedge f_1$, $f = f_0 \times f_1$, or $f = f_0 \sqcap f_1$, etc.

Which one is better? Does anyone have any better idea?

Answer 1

What you are describing is the canonical map from the coproduct (in the category of sets and functions). Standard notation for this is $[f_0,f_1]$.

Answer 2

$f$ cannot be $f_0 \times f_1$ since this function would take 2 arguments. What you want is $f_0 \sqcup f_1$, the disjoint union of the 2 functions, acting on the disjoint union of the 2 sets: $X_0 \sqcup X_1$.