Function composition

If f₁: X → Y and f₂: X' → Y', and Y is a subset of X', then the composition of f₁ and f₂ is the function f₃: X → Y' such that

for all x ∈ X, f₃(x) = f₂(f₁(x))

The function f₃ has the same effect as first applying f₁, then applying f₂, as shown below:

The function f₃ is written f₃ = f₂ · f₁
Note that f₁ is applied first, despite being to the right of f₂.
If the domain and the range of a function are the same, i.e. f: X → X, then f² = f · f, f³ = f² · f, …

Recall that a modem transmitter is a function

ModemTransmitter: BitSequences → Sounds.

The telephone network is a function

TelephoneNetwork: Sounds → Sounds.

A modem receiver is a function

ModemReceiver: Sounds → BitSequences.

Ideally, a modem design is a function such that

ModemReceiver · TelephoneNetwork · ModemTransmitter = Identity

where Identity: BitSequences → BitSequences is an identity function, where

for all x ∈ BitSequences, Identity(x) = x.

The challenge, of course, is to come up with a design that satisfies this property for a wide range of functions TelephoneNetwork.

Week 2