Set theory

From Wikimedia Commons, the free media repository
Jump to navigation Jump to search
English: Set theory is a branch of Mathematics.
It's regarded the foundation of mathematics, and closely related with logic.


Operations on and relations between two sets

[edit]

The Venn diagrams in the left matrix represent set operations - e.g. the intersection ,
those in the right matrix represent set relations - e.g. the subset relation , more usually represented by an Euler diagram:
The set theoretic descriptions are over the Venn diagrams:

 
c
          
A = A
1111 1111
 
Ac  Bc
true
A ↔ A
 
 B
 
 Bc
AA
 
 
 Bc
1110 0111 1110 0111
 
 Bc
¬A  ¬B
A → ¬B
 
 B
 B
A ← ¬B
 
Ac B
 
A B
A¬B
 
 
A = Bc
A¬B
 
 
A B
1101 0110 1011 1101 0110 1011
 
Bc
 ¬B
A ← B
 
A
 B
A ↔ ¬B
 
Ac
¬A  B
A → B
 
B
 
B =
AB
 
 
A = c
A¬B
 
 
A =
AB
 
 
B = c
1100 0101 1010 0011 1100 0101 1010 0011
¬B
 
 
 Bc
A
 
 
(A  B)c
¬A
 
 
Ac  B
B
 
Bfalse
 
Atrue
 
 
A = B
Afalse
 
Btrue
 
0100 1001 0010 0100 1001 0010
 ¬B
 
 
Ac  Bc
 B
 
 
 B
¬A  B
 
AB
 
1000 0001 1000 0001
¬A  ¬B
 
 
 B
 
 
A = Ac
0000 0000
false
A ↔ ¬A
A¬A
 
These sets (statements) have complements (negations).
They are in the opposite position within this matrix.
These relations are statements, and have negations.
They are shown in a separate matrix in the box below.


Syllogisms

[edit]

Syllogisms can be described in the language of set theory.

1
Barbara

Barbari

Darii

Ferio

Celaront

Celarent
2
Festino

Cesaro

Cesare

Camestres

Camestros

Baroco
3
Darapti

Datisi

Disamis

Felapton

Ferison

Bocardo
4
Bamalip

Dimatis

Fesapo

Fresison

Calemes

Calemos
Venn- and Euler diagrams

Partitions

[edit]

Various files

[edit]