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Good-to-know Stuff

Set theory

Theorem
  1. \(A\cup B=A\) if and only if \(B\subset A\)
  2. \(A\cap B=A\) if and only if \(A\subset B\)
  3. \(A-B=A\) if and only if \(A\cap B=\varnothing\)
  4. \(A\cup B\supset A\) and \(A\cup B\supset B\)
  5. \(A\cap B\subset A\) and \(A\cap B\subset B\)
  6. \(\varnothing\subset A\) for any set \(A\).
  7. \(A-B=\varnothing\) if and only if \(A=B\)
  8. \(A=B\) if and only if \(A\cup B=A\cap B\)
  9. \(\varnothing\cup A=A\) and \(\varnothing\cap A=\varnothing\) for any set \(A\).