Sets

A collection of distinct objects, which are called elements or points of the set

Terminology

• $S$ is the universal set (the collection of all points of interest in the current situation)
• $A\subset B$ subset
• $A\subseteq B$ improper subset
• $\varnothing$ null or empty sett
• Mutually exclusive/disjoint: $A\cap B=\varnothing$

Operations

• Union: $A\cup B$ (elements in either)
• Intersect: $A\cap B$ (elements in both)
• Difference: $A\backslash B$ (elements in first but not second)
• Complement: $\overline{A}$ (collection of all points in $S$ that are not in $A$)
• Power set: $2^A$ (set of all subsets)

Laws

• Distributive
  • $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
  • $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
• DeMorgan’s
  • $\overline{A\cap B}=\overline{A}\cup \overline{B}$
  • $\overline{A\cup B}=\overline{A}\cap \overline{B}$