The Logic of Subsets and Supersets

$A \cup (B \cap C) = (A \cup B) \cap C$

$A \cup (B \cup C) = (A \cup B) \cup C$

$A \cap (B \cap C) = (A \cap B) \cap C$

$A \cap B \subseteq C$ if and only if $A \subseteq C$ and $B \subseteq C$

$A \cap B \subseteq C$ if and only if $A \cup B \subseteq C$

$A \subseteq B$ and $B \subseteq C$ implies $A \subseteq C$

$A \subseteq B$ and $B \subseteq A \cup C$ implies $A \subseteq B \cap C$