Function composition
If f_{1}: X → Y and f_{2}: X' → Y', and Y is a subset of X', then the composition of f_{1} and f_{2} is the function f_{3}: X → Y' such that
The function f_{3} has the same effect as first applying f_{1}, then applying f_{2}, as shown below:
- The function f_{3} is written f_{3} = f_{2} · f_{1}
- Note that f_{1} is applied first, despite being to the right of f_{2}.
- If the domain and the range of a function are the same, i.e. f: X → X, then f^{ 2} = f · f, f^{ 3} = f^{ 2} · f, …
The telephone network is a function
A modem receiver is a function
Ideally, a modem design is a function such that
where Identity: BitSequences → BitSequences is an identity function, where
The challenge, of course, is to come up with a design that satisfies this property for a wide range of functions TelephoneNetwork.