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

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Partitions
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of December 6, 2017

2. Definition
Let X be a family of sets.
X is pairwise disjoint if every two different sets in X are disjoint.
Definition Let X be a set.
A partition of X is a family P of sets with the following properties.
Cover: P = X,
Disjointness: P is a pairwise disjoint family of sets,
Non-empty components: ∅ ∈ P.

3. Example
For i = 0, 1, 2, let Xi = {n : ∃k∈Z n = 3k + i}.
The family {X0, X1, X2} is a partition of Z.
Exercise
1. List all the partitions of {1, 2, 3}.
2. List all the partitions of {1, 2}.
3. List all the partitions of {1}.
4. List all the partitions of ∅.