notebook/notes/algebra/set.md

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title TARGET DECK FILE TAGS tags
Algebra of Sets Obsidian::STEM algebra::set set
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%%

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.

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: A non-empty set. 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

The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.

%%ANKI Cloze {\cup} is to algebra of sets whereas {+} is to algebra of real numbers. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Cloze {\cap} is to algebra of sets whereas {\cdot} is to algebra of real numbers. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Cloze {-} is to algebra of sets whereas {-} is to algebra of real numbers. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

%%ANKI Cloze {\subseteq} is to algebra of sets whereas {\leq} is to algebra of real numbers. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

END%%

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.

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.

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.

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.

END%%

%%ANKI Basic Under what two conditions is A \times B = B \times A? Back: A = B or either set is the empty set. Reference: “Cartesian Product,” in Wikipedia, April 17, 2024, 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.

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.

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 distributive 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.

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.

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.

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.

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.

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.

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.

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 e.g. 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.

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 functions 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 any function F \colon A \rightarrow B, F is a member of what other set? Back: \mathscr{P}(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).