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Mathematics LibreTexts

3.4: Summary

  • Page ID
    23894
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    • Important definitions:
      • set
      • element
      • subset
      • proper subset
      • predicate
      • union
      • intersection
      • set difference
      • complement
      • disjoint
      • pairwise-disjoint
      • power set
    • A set is unordered and without repetition.
    • \(\emptyset\) and \(A\) are subsets of \(A\).
    • \(A = B\) if and only if we have both \(A \subset B\) and \(B \subset A\).
    • For our purposes, predicates usually have only one or two variables.
    • If a predicate has two variables, the order of the variables is important.
    • Venn diagrams are a tool for illustrating set operations.
    • \(\#\mathcal{P}(A) = 2^{\#A}\)
    • Notation:
      • \(\{ \ \ \ \}\)
      • \(\in\), \(\notin\)
      • \(\emptyset\) (empty set)
      • \(\#A\)
      • \(A \subset B\), \(A \not\subset B\), \(A \supset B\)
      • \(P(x)\), \(x \mathrel{Q} y\) (predicates)
      • \(\{\, a \in A \mid P(a) \,\}\)
      • \(\mathcal{U}\) (universe of discourse)
      • \(\mathbb{N}\), \(\mathbb{N}^+\), \(\mathbb{Z}\), \(\mathbb{Q}\), \(\mathbb{R}\)
      • \(A \cup B\)
      • \(A \cap B\)
      • \(A \setminus B\)
      • \(\bar{A}\)
      • \(\mathcal{P}(A)\)
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