Set theory

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

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

Ac $\scriptstyle \cup$ Bc

true A ↔ A

A $\scriptstyle \cup$ B

A $\scriptstyle \subseteq$ Bc

A$\scriptstyle \Leftrightarrow$A

A $\scriptstyle \supseteq$ Bc

A $\scriptstyle \cup$ Bc

¬A $\scriptstyle \lor$ ¬B A → ¬B

A $\scriptstyle \Delta$ B

A $\scriptstyle \lor$ B A ← ¬B

Ac $\scriptstyle \cup$ B

A $\scriptstyle \supseteq$ B

A$\scriptstyle \Rightarrow$¬B

A = Bc

A$\scriptstyle \Leftarrow$¬B

A $\scriptstyle \subseteq$ B

Bc

A $\scriptstyle \lor$ ¬B A ← B

A

A $\scriptstyle \oplus$ B A ↔ ¬B

Ac

¬A $\scriptstyle \lor$ B A → B

B

B = ∅

A$\scriptstyle \Leftarrow$B

A = ∅c

A$\scriptstyle \Leftrightarrow$¬B

A = ∅

A$\scriptstyle \Rightarrow$B

B = ∅c

¬B

A $\scriptstyle \cap$ Bc

A

(A $\scriptstyle \Delta$ B)c

¬A

Ac $\scriptstyle \cap$ B

B

B$\scriptstyle \Leftrightarrow$false

A$\scriptstyle \Leftrightarrow$true

A = B

A$\scriptstyle \Leftrightarrow$false

B$\scriptstyle \Leftrightarrow$true

A $\scriptstyle \land$ ¬B

Ac $\scriptstyle \cap$ Bc

A $\scriptstyle \leftrightarrow$ B

A $\scriptstyle \cap$ B

¬A $\scriptstyle \land$ B

A$\scriptstyle \Leftrightarrow$B

¬A $\scriptstyle \land$ ¬B

∅

A $\scriptstyle \land$ B

A = Ac

false A ↔ ¬A

A$\scriptstyle \Leftrightarrow$¬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.

more relations

The operations, arranged in the same matrix as above. The 2x2 matrices show the same information like the Venn diagrams. (This matrix is similar to this Hasse diagram.)    In set theory the Venn diagrams represent the set, which is marked in red.   These 15 relations, except the empty one, are minterms and can be the case. The relations in the files below are disjunctions. The red fields of their 4x4 matrices tell, in which of these cases the relation is true. (Inherently only conjunctions can be the case. Disjunctions are true in several cases.) In set theory the Venn diagrams tell, that there is an element in every red, and there is no element in any black intersection. Negations of the relations in the matrix on the right. In the Venn diagrams the negation exchanges black and red.   In set theory the Venn diagrams tell, that there is an element in one of the red intersections. (The existential quantifications for the red intersections are combined by or. They can be combined by the exclusive or as well.) Relations like subset and implication, arranged in the same kind of matrix as above.   In set theory the Venn diagrams tell, that there is no element in any black intersection.

Syllogisms

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

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Various files

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  Venn 0101 0101.svg
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  Venn 0011 0011.svg
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  Venn 0000 1111.svg
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  Venn 0000 0001.svg
  All 256 Venn diagrams of this kind can be found here

 

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