--- title: Algebra of Sets TARGET DECK: Obsidian::STEM FILE TAGS: algebra::set set tags: - algebra - set --- ## Overview The study of the operations of union ($\cup$), intersection ($\cap$), and set difference ($-$), together with the inclusion relation ($\subseteq$), goes by the **algebra of sets**. %%ANKI Basic What three operators make up the algebra of sets? Back: $\cup$, $\cap$, and $-$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What *relation* is relevant in the algebra of sets? Back: $\subseteq$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Symmetric Difference Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B) \cup (B - A)$$ %%ANKI Basic What two operators are used in the definition of the symmetric difference? Back: $\cup$ and $-$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the symmetric difference of sets $A$ and $B$ denoted? Back: $A \mathop{\triangle} B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $A \mathop{\triangle} B$ defined? Back: As $(A - B) \cup (B - A)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is the symmetric difference commutative? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is the symmetric difference associative? Back: Yes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Cartesian Product Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$ %%ANKI Basic How is the Cartesian product of $A$ and $B$ denoted? Back: $A \times B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Using ordered pairs, how is $A \times B$ defined? Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Who is attributed the representation of points in a plane? Back: René Descartes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why is the Cartesian product named the way it is? Back: It is named after René Descartes. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic Suppose $x, y \in A$. What set, derived from $A$, is $\langle x, y \rangle$ a member of? Back: $\mathscr{P}\mathscr{P}A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% We can also form (something like) the Cartesian product of infinitely many sets, provided that the sets are suitably indexed. Let $I$ be an index set and $H$ a function whose domain includes $I$. Define $$\bigtimes_{i \in I} H(i) = \{f \mid f \text{ is a function with domain } I \text{ and } \forall i \in I, f(i) \in H(i)\}$$ %%ANKI Basic What kind of mathematical object is $I$ in $\bigtimes_{i \in I} H(i)$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $H$ in $\bigtimes_{i \in I} H(i)$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the domain of $H$ in $\bigtimes_{i \in I} H(i)$? Back: Some superset of $I$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the range of $H$ in $\bigtimes_{i \in I} H(i)$? Back: Some superset of $\{H(i) \mid i \in I\}$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $I$ be an index set and $H$ a function such that $I \subseteq \mathop{\text{dom}}H$. How is $\bigtimes_{i \in I} H(i)$ defined? Back: $\bigtimes_{i \in I} H(i) = \{ f \mid f \text{ is a function with domain } I \text { and } \forall i \in I, f(i) \in H(i) \}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $h \in \bigtimes_{i \in I} H(i)$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \in \bigtimes_{i \in I} H(i)$. What is the domain of $f$? Back: $I$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $f \in \bigtimes_{i \in I} H(i)$. What is the codomain of $f$? Back: $\bigcup_{i \in I} H(i)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given arbitrary sets $A$ and $B$, what index set $I$ and function $H$ satisfies $A \times B = \bigtimes_{i \in I} H(i)$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* can't $A \times B$ be rewritten with $\bigtimes_{i \in I} H(i)$ assuming suitable $I$ and $H$? Back: The former is a set of ordered pairs. The latter is a set of functions. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Assume AoC and $H(j) = \varnothing$ for some $j \in I$. What does $\bigtimes_{i \in I} H(i)$ evaluate to? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic When does $\bigtimes_{i \in I} H(i) = \varnothing$? Back: When there exists some $i \in I$ such that $H(i) = \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Assume AoC and $H(j) \neq \varnothing$ for all $j \in I$. What does $\bigtimes_{i \in I} H(i)$ evaluate to? Back: $\{f \mid f \text{ is a function with domain } I \text{ and } \forall i \in I, f(i) \in H(i)\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following is likely a diagram of what? ![[infinite-cartesian-product.png]] Back: A member of $\bigtimes_{i \in \omega} H(i)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $H(i) \neq \varnothing$ for all $i \in I$. When is $\bigtimes_{i \in I} H(i) \neq \varnothing$? Back: When AoC is included in our formal system. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Laws ### Commutative Laws For any sets $A$ and $B$, $$\begin{align*} A \cup B & = B \cup A \\ A \cap B & = B \cap A \end{align*}$$ %%ANKI Basic The commutative laws of the algebra of sets apply to what operators? Back: $\cup$ and $\cap$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the union commutative law state? Back: For any sets $A$ and $B$, $A \cup B = B \cup A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the intersection commutative law state? Back: For any sets $A$ and $B$, $A \cap B = B \cap A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is the Cartesian product commutative? Back: No. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic *Why* isn't the Cartesian product commutative? Back: Because the Cartesian product comprises of *ordered* pairs. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic Suppose $A \neq \varnothing$ and $B \neq \varnothing$. When does $A \times B = B \times A$? Back: When $A = B$. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic Suppose $A \neq \varnothing$ and $A \neq B$. When does $A \times B = B \times A$? Back: When $B = \varnothing$. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic Under what three conditions is $A \times B = B \times A$? Back: $A = B$ or $A = \varnothing$ or $B = \varnothing$. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% ### Associative Laws For any sets $A$ and $B$, $$\begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}$$ %%ANKI Basic The associative laws of the algebra of sets apply to what operators? Back: $\cup$ and $\cap$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the union associative law state? Back: For any sets $A$, $B$, and $C$, $A \cup (B \cup C) = (A \cup B) \cup C$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the intersection associative law state? Back: For any sets $A$, $B$, and $C$, $A \cap (B \cap C) = (A \cap B) \cap C$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Is the Cartesian product associative? Back: No. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic *Why* isn't the Cartesian product associative? Back: The association of parentheses defines the nesting of the ordered pairs. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% ### Distributive Laws For any sets $A$, $B$, and $C$, $$\begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}$$ %%ANKI Basic The distributive laws of the algebra of sets apply to what operators? Back: $\cup$ and $\cap$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze The distributive law states {$A \cap (B \cup C)$} $=$ {$(A \cap B) \cup (A \cap C)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze The distributive law states {$A \cup (B \cap C)$} $=$ {$(A \cup B) \cap (A \cup C)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What concept in set theory relates the algebra of sets to boolean algebra? Back: Membership. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What two equalities relates $A \cup B$ with $a \lor b$? Back: $a = (x \in A)$ and $b = (x \in B)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What two equalities relates $A \cap B$ with $a \land b$? Back: $a = (x \in A)$ and $b = (x \in B)$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% More generally, for any sets $A$ and $\mathscr{B}$, $$\begin{align*} A \cup \bigcap \mathscr{B} & = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}, \text{ for } \mathscr{B} \neq \varnothing \\ A \cap \bigcup \mathscr{B} & = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\} \end{align*}$$ %%ANKI Basic What is the generalization of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$? Back: $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of identity $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$? Back: $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ for $\mathscr{B} \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Assuming $\mathscr{B} \neq \varnothing$, the distributive law states {$A \cup \bigcap \mathscr{B}$} $=$ {$\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze The distributive law states {$A \cap \bigcup \mathscr{B}$} $=$ {$\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{A \cup X \mid X \in \mathscr{B}\}$ pronounced? Back: The set of all $A \cup X$ such that $X \in \mathscr{B}$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $A \cap \bigcup \mathscr{B} = \bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$? Back: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $A \cup \bigcap \mathscr{B} = \bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$? Back: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets larger? Back: Larger. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcup\, \{A \cap X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B}$ gets smaller? Back: Smaller. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets larger? Back: Smaller. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcap\, \{A \cup X \mid X \in \mathscr{B}\}$ get smaller or larger as $\mathscr{B} \neq \varnothing$ gets smaller? Back: Larger. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% For any sets $A$, $B$, and $C$, $$\begin{align*} A \times (B \cap C) & = (A \times B) \cap (A \times C) \\ A \times (B \cup C) & = (A \times B) \cup (A \times C) \\ A \times (B - C) & = (A \times B) - (A \times C) \end{align*}$$ %%ANKI Basic Which of the algebra of sets operators does the Cartesian product distribute over? Back: $\cap$, $\cup$, and $-$ Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic What distributivity rule is satisfied by $\cap$ and $\times$? Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$ Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Cloze The Cartesian product satisfies distributivity: {$A \times (B \cap C)$} $=$ {$(A \times B) \cap (A \times C)$}. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic What distributivity rule is satisfied by $\cup$ and $\times$? Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$ Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Cloze The Cartesian product satisfies distributivity: {$A \times (B \cup C)$} $=$ {$(A \times B) \cup (A \times C)$}. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Basic What distributivity rule is satisfied by $-$ and $\times$? Back: $A \times (B - C) = (A \times B) - (A \times C)$ Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% %%ANKI Cloze The Cartesian product satisfies distributivity: {$A \times (B - C)$} $=$ {$(A \times B) - (A \times C)$}. Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% In addition, $$\begin{align*} A \times \bigcup \mathscr{B} & = \bigcup\, \{A \times X \mid X \in \mathscr{B}\} \\ A \times \bigcap \mathscr{B} & = \bigcap\, \{A \times X \mid X \in \mathscr{B}\} \end{align*}$$ %%ANKI Basic What is the generalization of identity $A \times (B \cup C) = (A \times B) \cup (A \times C)$? Back: $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$? Back: $A \times (B \cap C) = (A \times B) \cap (A \times C)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of identity $A \times (B \cap C) = (A \times B) \cap (A \times C)$? Back: $A \times \bigcap \mathscr{B} = \bigcap\, \{A \times X \mid X \in \mathscr{B}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $A \times \bigcup \mathscr{B} = \bigcup\, \{A \times X \mid X \in \mathscr{B}\}$? Back: $A \times (B \cup C) = (A \times B) \cup (A \times C)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### De Morgan's Laws For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$ %%ANKI Basic The De Morgan's laws of the algebra of sets apply to what operators? Back: $\cup$, $\cap$, and $-$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze De Morgan's law states that {$C - (A \cup B)$} $=$ {$(C - A) \cap (C - B)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze De Morgan's law states that {$C - (A \cap B)$} $=$ {$(C - A) \cup (C - B)$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For their respective De Morgan's laws, {$-$} is to the algebra of sets whereas {$\neg$} is to boolean algebra. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For their respective De Morgan's laws, {$\cup$} is to the algebra of sets whereas {$\lor$} is to boolean algebra. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For their respective De Morgan's laws, {$\cap$} is to the algebra of sets whereas {$\land$} is to boolean algebra. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% More generally, for any sets $C$ and $\mathscr{A} \neq \varnothing$, $$\begin{align*} C - \bigcup \mathscr{A} & = \bigcap\, \{C - X \mid X \in \mathscr{A}\} \\ C - \bigcap \mathscr{A} & = \bigcup\, \{C - X \mid X \in \mathscr{A}\} \end{align*}$$ %%ANKI Basic What is the generalization of identity $C - (A \cup B) = (C - A) \cap (C - B)$? Back: $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the generalization of identity $C - (A \cap B) = (C - A) \cup (C - B)$? Back: $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcap \mathscr{A}$} $=$ {$\bigcup\, \{C - X \mid X \in \mathscr{A}\}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$? Back: $C - (A \cup B) = (C - A) \cap (C - B)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the specialization of identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$? Back: $C - (A \cap B) = (C - A) \cup (C - B)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which law of the algebra of sets is represented by the following? $$C - (A \cup B) = (C - A) \cap (C - B)$$ Back: De Morgan's Law. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze For $\mathscr{A} \neq \varnothing$, De Morgan's law states that {$C - \bigcup \mathscr{A}$} $=$ {$\bigcap\, \{C - X \mid X \in \mathscr{A}\}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why does identity $C - \bigcup \mathscr{A} = \bigcap\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$? Back: The RHS evaluates to class $\bigcap \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why does identity $C - \bigcap \mathscr{A} = \bigcup\, \{C - X \mid X \in \mathscr{A}\}$ fail when $\mathscr{A} = \varnothing$? Back: $\bigcap \mathscr{A}$ is undefined. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger? Back: Smaller. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcap\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller? Back: Larger. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets larger? Back: Larger. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Does $\bigcup\, \{C - X \mid X \in \mathscr{A}\}$ get smaller or larger as $\mathscr{A} \neq \varnothing$ gets smaller? Back: Smaller. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Monotonicity Let $A$, $B$, and $C$ be arbitrary sets. Then * $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$, * $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$, * $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$ %%ANKI Basic The monotonicity properties of $\subseteq$ are what kind of propositional logical statement? Back: Implications. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the shared antecedent of the monotonicity properties of $\subseteq$? Back: $A \subseteq B$ for some sets $A$ and $B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cup$ operator. Back: $A \subseteq B \Rightarrow A \cup C \subseteq B \cup C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$, $B$, and $C$, state the monotonicity property of $\subseteq$ related to the $\cap$ operator. Back: $A \subseteq B \Rightarrow A \cap C \subseteq B \cap C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$ and $B$, state the monotonicity property of $\subseteq$ related to the $\bigcup$ operator. Back: $A \subseteq B \Rightarrow \bigcup A \subseteq \bigcup B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why are the monotonicity properties of $\subseteq$ named the way they are? Back: The ordering of operands in the antecedent are preserved in the consequent. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% In addition, * $A \subseteq B \Rightarrow A \times C \subseteq B \times C$ %%ANKI Basic What monotonicity property does the Cartesian product satisfy? Back: $A \subseteq B \Rightarrow A \times C \subseteq B \times C$ Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305). END%% ### Antimonotonicity Let $A$, $B$, and $C$ be arbitrary sets. Then * $A \subseteq B \Rightarrow C - B \subseteq C - A$, * $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$ %%ANKI Basic What kind of propositional logical statements are the antimonotonicity properties of $\subseteq$? Back: Implications. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the shared antecedent of the antimonotonicity properties of $\subseteq$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {1:Monotonicity} of $\subseteq$ is to {2:$\bigcup$} whereas {2:antimonotonicity} of $\subseteq$ is to {1:$\bigcap$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why are the antimonotonicity properties of $\subseteq$ named the way they are? Back: The ordering of operands in the antecedent are reversed in the consequent. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$ and $B$, state the antimonotonicity property of $\subseteq$ related to the $\bigcap$ operator. Back: $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$, $B$, and $C$, state the antimonotonicity property of $\subseteq$ related to the $-$ operator. Back: $A \subseteq B \Rightarrow C - B \subseteq C - A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why do we need the empty set check in $\varnothing \neq A \subseteq B \Rightarrow \bigcap B \subseteq \bigcap A$? Back: $\bigcap A$ is not a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Cancellation Laws Let $A$, $B$, and $C$ be sets. If $A \neq \varnothing$, * $(A \times B = A \times C) \Rightarrow B = C$ * $(B \times A = C \times A) \Rightarrow B = C$ %%ANKI Basic What is the left cancellation law of the Cartesian product? Back: If $A \neq \varnothing$ then $(A \times B = A \times C) \Rightarrow B = C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic $(A \times B = A \times C) \Rightarrow B = C$ is always true if what condition is satisfied? Back: $A \neq \varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the right cancellation law of the Cartesian product? Back: If $A \neq \varnothing$ then $(B \times A = C \times A) \Rightarrow B = C$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic $(B \times A = C \times A) \Rightarrow B = C$ is always true if what condition is satisfied? Back: $A \neq \varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Index Sets Let $I$ be a set, called the **index set**. Let $F$ be a [[set/functions|function]] whose domain includes $I$. Then we define $$\bigcup_{i \in I} F(i) = \bigcup\,\{F(i) \mid i \in I\}$$ and, if $I \neq \varnothing$, $$\bigcap_{i \in I} F(i) = \bigcap\, \{F(i) \mid i \in I\}$$ %%ANKI Basic What name does $I$ go by in expression $\bigcup_{i \in I} F(i)$? Back: The "index set". Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcup_{i \in I} F(i)$ alternatively denoted? Back: $\bigcup_{i \in I} F_i$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $I$ in expression $\bigcup_{i \in I} F(i)$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $F$ in expression $\bigcup_{i \in I} F(i)$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcup_{i \in I} F_i$ alternatively denoted? Back: $\bigcup_{i \in I} F(i)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $F$ in expression $\bigcup_{i \in I} F_i$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the domain of $F$ assumed to be in expression $\bigcup_{i \in I} F(i)$? Back: Some superset of $I$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition must $I$ satisfy in expression $\bigcup_{i \in I} F(i)$? Back: N/A. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $I = \{0, 1, 2\}$. What does $\bigcup_{i \in I} F(i)$ evaluate to? Back: $F(0) \cup F(1) \cup F(2)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $I = \varnothing$. What does $\bigcup_{i \in I} F(i)$ evaluate to? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $F$ in expression $\bigcap_{i \in I} F(i)$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcap_{i \in I} F(i)$ often alternatively denoted? Back: $\bigcap_{i \in I} F_i$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the domain of $F$ assumed to be in expression $\bigcap_{i \in I} F(i)$? Back: Some superset of $I$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What condition must $I$ satisfy in expression $\bigcap_{i \in I} F(i)$? Back: $I \neq \varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $I = \{0, 1, 2\}$. What does $\bigcap_{i \in I} F(i)$ evaluate to? Back: $F(0) \cap F(1) \cap F(2)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Suppose $I = \varnothing$. What does $\bigcap_{i \in I} F(i)$ evaluate to? Back: N/A. This is undefined. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcap_{i \in I} F_i$ alternatively denoted? Back: $\bigcap_{i \in I} F(i)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is $F$ in expression $\bigcap_{i \in I} F_i$? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Function Sets For sets $A$ and $B$, the collection of functions $F$ from $A$ into $B$ is: $$^AB = \{F \mid F \colon A \rightarrow B\}$$ $^AB$ is read as "$B$-pre-$A$". It is often written as $B^A$ instead. %%ANKI Basic For sets $A$ and $B$, how is set $B^A$ defined? Back: $\{F \mid F \colon A \rightarrow B\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For sets $A$ and $B$, how is set $^AB$ defined? Back: $\{F \mid F \colon A \rightarrow B\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For any function $F \colon A \rightarrow B$, $F$ is a subset of what other set? Back: $A \times B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For sets $A$ and $B$, how is set $B^A$ pronounced? Back: As "$B$-pre-$A$". Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why prefer notation $B^A$ over $^AB$? Back: The notation mirrors $|B|^{|A|}$, the number of elements in $B^A$ given both sets are finite. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For sets $A$ and $B$, how is set $^AB$ pronounced? Back: As "$B$-pre-$A$". Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why prefer notation $^AB$ over $B^A$? Back: Because the sets are written left-to-right, from domain to codomain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* is set $B^A$ denoted the way it is? Back: If $A$ and $B$ are finite, then $B^A$ has $|B|^{|A|}$ members. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the domain of $^\omega\{0, 1\}$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the domain of a member of $^\omega\{0, 1\}$? Back: $\omega$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the range of $\{0, 1\}^\omega$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the range of a member of $\{0, 1\}^\omega$? Back: $\{0, 1\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does $\varnothing^\varnothing$ evaluate to? Back: $\{\varnothing\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For $A \neq \varnothing$, what does $\varnothing^A$ evaluate to? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For $A \neq \varnothing$, *why* does $\varnothing^A = \varnothing$? Back: No function can map a nonempty domain to an empty range. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For $A \neq \varnothing$, what does $^\varnothing A$ evaluate to? Back: $\{\varnothing\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For $A \neq \varnothing$, *why* does $^\varnothing A = \{\varnothing\}$? Back: $\varnothing$ is the only function with empty domain. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Bibliography * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).