5.S: Set Theory (Summary)

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.