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