# 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 ${\displaystyle \scriptstyle \cup }$ Bc trueA ↔ A A ${\displaystyle \scriptstyle \cup }$ B A ${\displaystyle \scriptstyle \subseteq }$ Bc A${\displaystyle \scriptstyle \Leftrightarrow }$A A ${\displaystyle \scriptstyle \supseteq }$ Bc A ${\displaystyle \scriptstyle \cup }$ Bc ¬A ${\displaystyle \scriptstyle \lor }$ ¬BA → ¬B A ${\displaystyle \scriptstyle \Delta }$ B A ${\displaystyle \scriptstyle \lor }$ BA ← ¬B Ac ${\displaystyle \scriptstyle \cup }$ B A ${\displaystyle \scriptstyle \supseteq }$ B A${\displaystyle \scriptstyle \Rightarrow }$¬B A = Bc A${\displaystyle \scriptstyle \Leftarrow }$¬B A ${\displaystyle \scriptstyle \subseteq }$ B Bc A ${\displaystyle \scriptstyle \lor }$ ¬BA ← B A A ${\displaystyle \scriptstyle \oplus }$ BA ↔ ¬B Ac ¬A ${\displaystyle \scriptstyle \lor }$ BA → B B B = ∅ A${\displaystyle \scriptstyle \Leftarrow }$B A = ∅c A${\displaystyle \scriptstyle \Leftrightarrow }$¬B A = ∅ A${\displaystyle \scriptstyle \Rightarrow }$B B = ∅c ¬B A ${\displaystyle \scriptstyle \cap }$ Bc A (A ${\displaystyle \scriptstyle \Delta }$ B)c ¬A Ac ${\displaystyle \scriptstyle \cap }$ B B B${\displaystyle \scriptstyle \Leftrightarrow }$false A${\displaystyle \scriptstyle \Leftrightarrow }$true A = B A${\displaystyle \scriptstyle \Leftrightarrow }$false B${\displaystyle \scriptstyle \Leftrightarrow }$true A ${\displaystyle \scriptstyle \land }$ ¬B Ac ${\displaystyle \scriptstyle \cap }$ Bc A ${\displaystyle \scriptstyle \leftrightarrow }$ B A ${\displaystyle \scriptstyle \cap }$ B ¬A ${\displaystyle \scriptstyle \land }$ B A${\displaystyle \scriptstyle \Leftrightarrow }$B ¬A ${\displaystyle \scriptstyle \land }$ ¬B ∅ A ${\displaystyle \scriptstyle \land }$ B A = Ac falseA ↔ ¬A A${\displaystyle \scriptstyle \Leftrightarrow }$¬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.

## 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