Total Maps (Functions)

Lemma: Composition of Total Functions

Notation

\((g\circ f)(x)\)

\((g(f(x))\)

Let \(A,B,C,D\) be sets, let \(B\) be a subset of \(C\) and let \(f:A\mapsto B\) and \(g:C\mapsto D\) be two total functions. Then the composition of the functions
\[(g\circ f)(a):=\cases{A&\(\mapsto D\)\\
a&\(\mapsto g(f(a))\)}\]

is also a total function.

References

[577] Knauer Ulrich: “Diskrete Strukturen - kurz gefasst”, Spektrum Akademischer Verlag, 2001