Definition:Closure Axioms (Matroid)

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Let $S$ be a finite set.

Let $\sigma : \powerset S \to \powerset S$ be a mapping where $\powerset S$ Denotes the power set of $S$.

$\sigma$ is said to satisfy the closure axioms if:

\((S1)\)   $:$     \(\displaystyle \forall X \in \powerset S:\) \(\displaystyle X \subseteq \map \sigma X \)             
\((S2)\)   $:$     \(\displaystyle \forall X, Y \in \powerset S:\) \(\displaystyle X \subseteq Y \implies \map \sigma X \subseteq \map\sigma Y \)             
\((S3)\)   $:$     \(\displaystyle \forall X \in \powerset S:\) \(\displaystyle \map \sigma X = \map \sigma {\map \sigma X} \)             
\((S4)\)   $:$     \(\displaystyle \forall X \in \powerset S \land x, y \in S:\) \(\displaystyle y \not \in \map \sigma X \land y \in \map \sigma {X \cup \set x} \implies x \in \map \sigma {X \cup \set y} \)