5.S: Set Theory (Summary)
- Page ID
- 7065
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.