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Mathematics LibreTexts

5.S: Set Theory (Summary)

  • Page ID
    7065
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    Important Definitions

    • Equal sets, page 55

    • Subset, page 55

    • Proper subset, page 218

    • Power set, page 222

    • Cardinality of a finite set, page 223

    • Intersection of two sets, page 216

    • Union of two sets, page 216

    • Set difference, page 216

    • Complement of a set, page 216

    • Disjoint sets, page 236

    • Cartesian product of two sets, pages 256

    • Ordered pair, page 256

    • Union over a family of sets, page 265

    • Intersection over a family of sets, page 265

    • Indexing set, page 268

    • Indexed family of sets, page 268

    • Union over an indexed family of sets, page 269

    • Intersection over an indexed family of sets, page 269

    • Pairwise disjoint family of sets, page 272


    Important Theorems and Results about Sets

    • Theorem 5.5. Let \(n\) be a nonnegative integer and let \(A\) be a subset of some universal set. If \(A\) is a finite set with \(n\) elements, then \(A\) has \(2^n\) subsets. That is, if \(|A| = n\), then \(|\mathcal{P}(A)| = 2^n\). 
    • Theorem 5.18. Let \(A\), \(B\), and \(C\) be subsets of some universal set \(U\). Then all of the following equalities hold.

      Properties of the Empty Set               \(A \cap \emptyset = \emptyset\)                            \(A \cap U = A\)
      and the Universal Set                             \(A \cup \emptyset = A\)                           \(A \cup U = U\)

      Idempotent Laws                                      \(A \cap A = A\)                         \(A \cup A = A\)

      Commutative Laws.                                \(A \cap B = B \cap A\)                \(A \cup B = B \cup A\)

      Associative Laws                                      \((A \cap B) \cap C = A \cap (B \cap C)\)
                                                                                  \((A \cup B) \cup C = A \cup (B \cup C)\)

      Distributive Laws                                    \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)
                                                                                 \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\)
    • Theorem 5.20. Let \(A\) and \(B\) be subsets of some universal set \(U\). Then the following are true:
      \[\begin{array} {ll} {\text{Basic Properties}} & & {(A^c)^c = A} \\ {} & & {A - B = A \cap B^c} \\ {\text{Empty Set, Universal Set}\ \ \ \ \ \ \ \ \ \ \ \ \ } & & {A - \emptyset = A \text{ and } A - U = \emptyset} \\ {} & & {\emptyset ^c = U \text{ and } U^c = \emptyset} \\ {\text{De Morgan's Laws}} & & {(A \cap B)^c = A^c \cup B^c} \\ {} & & {(A \cup B)^c = A^c \cap B^c} \\ {\text{Subsets and Complements}} & & {A \subseteq B \text{ if and only if } B^c \subseteq A^c.} \end{array}\]
    • Theorem 5.25. Let \(A\), \(B\), and \(C\) be sets. Then

      1. \(A \times (B \cap C) = (A \times B) \cap (A \times C)\)
      2. \(A \times (B \cup C) = (A \times B) \cup (A \times C)\)
      3. \((A \cap B) \times C = (A \times C) \cap (B \times C)\)
      4. \((A \cup B) \times C = (A \times C) \cup (B \times C)\)
      5. \(A \times (B - C) = (A \times B) - (A \times C)\)
      6. \((A - B) \times C = (A \times C) - (B \times C)\)
      7. If \(T \subseteq A\), then \(T \times B \subseteq A \times B\).
      8. If \(T \subseteq B\), then \(A \times Y \subseteq A \times B\).
    • Theorem 5.30. Let \(\Lambda\) be a nonempty indexing set and let \(\mathcal{A} = \{A_{\alpha}\ |\ \alpha \in \Lambda\}\) be an indexed family of sets. Then

      1. For each \(\beta \in \Lambda\), \(\bigcap_{\alpha \in \Lambda}^{} A_{\alpha} \subseteq A_{\beta}\).
      2. For each \(\beta \in \Lambda\), \(A_{\beta} \subseteq \bigcap_{\alpha \in \Lambda}^{} A_{\alpha}\).
      3. \((\bigcap_{\alpha \in \Lambda}^{} A_{\alpha})^c = \bigcup_{\alpha \in \Lambda}^{} A_{\alpha} ^c\)
      4. \((\bigcup_{\alpha \in \Lambda}^{} A_{\alpha})^c = \bigcap_{\alpha \in \Lambda}^{} A_{\alpha} ^c\)

      Parts(3) and (4) are known as De Morgan's Laws.
    • Theorem 5.31. Let \(\Lambda\) be a nonempty indexing set, let \(\mathcal{A} = \{A_{\alpha}\ |\ \alpha \in \Lambda\}\) be an indexed family of sets, and let \(B\) be a set. Then

      1. \(B \cap (\bigcup_{\alpha \in \Lambda}^{} A_{\alpha}) = \bigcup_{\alpha \in \Lambda}^{} (B \cap A_{\alpha})\), and
      2. \(B \cup (\bigcap_{\alpha \in \Lambda}^{} A_{\alpha}) = \bigcap_{\alpha \in \Lambda}^{} (B \cup A_{\alpha})\),

    Important Proof Method

    The Choose-an-Element Method
    The choose-an-element method is frequently used when we encounter a universal quantifier in a statement in the backward process of a proof. This statement often has the form

    For each element with a given property, something happens.

    In the forward process of the proof, we then we choose an arbitrary element with the given property.

    Whenever we choose an arbitrary element with a given property, we are not selecting a specific element. Rather, the only thing we can assume about the element is the given property.

    For more information, see page 232.

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