Selected items from set theory and from methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Function powers
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of November 10, 2017

2. Definition
Let f : X −→ X. One defines recursively:
f0 = idX,
fn+1 = f ◦ fn, for any n ∈ N.
For any natural number n, fn is called the n-th function power of f or
the function power of f with the exponent n or the function power n of f .
Convention
We can drop the adjective “function” and say the n-th power of f
instead of “the n-th function power of f”.
Notes
Function powers are not defined for every f : X −→ Y ;
they are defined only if Y ⊆ X.
Power 0 is not defined for arbitrary binary relations.
Power 0 is defined for functions satisfying the condition above.

3. Informal Example
1. Let f(x) = 1.05x. This function gives the value of principal x after a year,
if deposited in a bank account that brings 5% annual interest.
Then, the power f10(x) is the total value after 10 years.
2. Let f be a function whose argument represents the atmospheric conditions, and
whose value represents the resulting atmospheric conditions 10 minutes later.
If x is an approximate current state of the atmosphere,
f24·6(x) is the state forecasted for 24 hours later.
(Chaos theory explains why
even a good approximation x of current conditions
makes fn(x), for high values of n,
a poor approximation of the actual future conditions.
So, long-term weather forecast is inherently unreliable.)

4. Proposition
Let f : X −→ X and m, n ∈ N. Then:
1. fm+n = fm ◦ fn = fn ◦ fm.
2. fm·n = (fm)n = (fn)m.
This can be proved by mathematical induction.
Exercise
Let f : X
1-1
−→ X.
1. Disprove: f1 = f2 ◦ f−1.
2. Disprove: (f0)−1 = (f−1)0.