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.