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
Union, intersection and complement
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 8, 2017

2. Union
Axioms imply:
∀X,Y ∃Z ∀u (u ∈ Z ↔ u ∈ X ∨ u ∈ Y ).
Definition
The union of X and Y , denoted X ∪ Y , is the set Z such that
∀u (u ∈ Z ↔ u ∈ X ∨ u ∈ Y ).

3. Union
X Y
Example
{1} ∪ {3} = {1, 3}.
{1, 2} ∪ {2, 3} = {1, 2, 3}.
{1, 2} ∪ {1, 2, 3} = {1, 2, 3}.

4. Union
Fact
1. z ∈ X ∪ Y iff z ∈ X or z ∈ Y .
2. z ∈ X ∪ Y iff z ∈ X and z ∈ Y .

5. Intersection
Definition
Let X and Y be any sets. The intersection of X and Y , denoted X ∩ Y , is the set
{z ∈ X : z ∈ Y }. (Think of it as: {z : z ∈ X ∧ z ∈ Y }.)
X Y
X ∩ Y
Example
{1, 2} ∩ {2, 3} = {2}.

6. Intersection
Fact
1. z ∈ X ∩ Y iff z ∈ X and z ∈ Y .
2. z ∈ X ∩ Y iff z ∈ X or z ∈ Y .

7. Complement
Definition
Let X ⊆ U. The complement of X with respect to U is the set
{u ∈ U : u ∈ X}. If U is known from the context it is denoted by Xc .
X Xc
U
Example
The complement of the set of even integers with respect to Z is the set of odd integers.
The complement of Q with respect to R is the set of all irrational numbers.

8. Complement
Fact
1. z ∈ Xc iff z ∈ X, for every z ∈ U.
2. z ∈ Xc iff z ∈ X, for every z ∈ U.

9. Implementation
1. A finite set of strings can be implemented as a dynamic hash table:
a string belongs to the set iff it is a key in the hash table.
Inserting, deleting, testing membership – average O(1) time.
Union, intersection, complement - in O(n) time).
2. Consider a universe U ={a0, ..., a7}.
Any subset A of U can be represented as a bit vector.
E.g. A={a0, a1, a3} is represented by the bit vector 11010000.
Bitwise operations |, &, ~ correspond to set operations ∪, ∩, c.
Very efficient; constant time (proportional to the size of U).

10. Exercises
1. ∅c =...
Uc =...
2. (Xc)c
=...

11. Exercises
True or false?
X ∪ Y =Y ∪ X
X ∩ Y =Y ∩ X

12. Exercises
Complete
1. X ∪ ∅=...
X ∩ U =...
2. X ∩ ∅=...
X ∪ U =...

13. Exercises
Complete
1. X ∪ X =...
X ∩ X =...
2. X ∪ Xc =...
X ∩ Xc =...

14. Exercises
Complete
1. X ⊆ Y iff X ∪ Y =...
X ⊇ Y iff X ∩ Y =...
2. X ⊆ Y iff Xc ∪ Y =...
X ⊇ Y iff Xc ∩ Y =...
3. X ⊆ Y iff Xc ∪ Y c =...
X ⊇ Y iff Xc ∩ Y c =...