set operations

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• Common set operations
  • Union: $S\cup T=\{x|x\in S\vee x\in T\}$: elements in S or T
    • $\vee$: or
  • Intersection: $S\cap T=\{x|x\in S,x\in T\}$: intersection of S and T only
  • Differences: $S-T=\{x|x\in S,x\notin T\}$: in S but not T
  • Complements: $\overline{S}=\{x|x\in \mathbb{U},x\notin S\}=\mathbb{U}-S$
    • $\mathbb{U}$ is the universe containing sets S and T
• Theorem 15
  • For all sets S and T, $S=(S\cap T)\cup (S-T)$
  • Prove Set Equality: show that $S\subset (S\cap T)\cup (S-T)$ and $(S\cap T)\cup (S-T)\subset S$
  • see the prove here