--- title: Set TARGET DECK: Obsidian::STEM FILE TAGS: set tags: - set --- ## Overview %%ANKI Basic How does Knuth define a *dynamic* set? Back: As a set that can change over time. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic How does Knuth distinguish mathematical sets from dynamic sets? Back: The former is assumed to be unchanging. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic How does Knuth define a dictionary? Back: As a dynamic set that allows insertions, deletions, and membership tests. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic Which of dynamic sets and dictionaries are more general? Back: The dynamic set. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic Is a dynamic set a dictionary? Back: Not necessarily. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic Is a dictionary a dynamic set? Back: Yes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Cloze A dictionary supports {insertions}, {deletions}, and {membership testing}. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). Tags: adt::dynamic_set END%% %%ANKI Basic Define the set of prime numbers less than $10$ using abstraction. Back: $\{x \mid x < 10 \land x \text{ is prime}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Define the set of prime numbers less than $5$ using set-builder notation. Back: $\{x \mid x < 5 \land x \text{ is prime}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Define the set of prime numbers less than $5$ using roster notation. Back: $\{2, 3\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Define the set of prime numbers less than $5$ using abstraction. Back: $\{x \mid x < 5 \land x \text{ is prime}\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What term describes the expression to the right of $\mid$ in set-builder notation? Back: The entrance requirement. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What term refers to $\_\_\; x\; \_\_$ in $\{x \mid \_\_\; x\; \_\_\}$? Back: The entrance requirement. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The term "entrance requirement" refers to what kind of set notation? Back: Set-builder/abstraction. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What name is given to set notation in which members are explicitly listed? Back: Roster notation. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does an atom refer to in set theory? Back: Any entity that is not a set but can exist in one. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze An {1:atom} in set theory is to {2:atomic} logical statements whereas {2:sets} are to {1:molecular} statements. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What distinguishes a set from an atom? Back: An atom cannot contain other entitites. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What intuition is broken when a box is viewed as an atom? Back: When viewed as an atom, the box is no longer a container. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Enderton's exposition makes what assumption about atoms? Back: The set of all atoms is the empty set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Extensionality If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ %%ANKI Basic What does the extensionality axiom state? Back: If two sets have exactly the same members, then they are equal. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the extensionality axiom expressed using first-order logic? Back: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ Back: The extensionality axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How many sets exist with no members? Back: Exactly one. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which set theory axiom proves uniqueness of $\varnothing$? Back: The extensionality axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Empty Set Axiom There exists a set having no members: $$\exists B, \forall x, x \not\in B$$ %%ANKI Basic What does the empty set axiom state? Back: There exists a set having no members. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the empty set axiom expressed using first-order logic? Back: $$\exists B, \forall x, x \not\in B$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\exists B, \forall x, x \not\in B$$ Back: The empty set axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which set theory axiom proves existence of $\varnothing$? Back: The empty set axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What two properties ensures definition $\varnothing$ is well-defined? Back: The empty set exists and is unique. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the empty set defined using set-builder notation? Back: $\{x \mid x \neq x\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Pairing Axiom For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$ %%ANKI Basic What does the pairing axiom state? Back: For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the pairing axiom expressed using first-order logic? Back: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$ Back: The pairing axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which set theory axiom proves existence of set $\{x, y\}$ where $x \neq y$? Back: The pairing axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which set theory axiom proves existence of set $\{x\}$? Back: The pairing axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic For sets $u$ and $v$, what name is given to set $\{u, v\}$? Back: The pair set of $u$ and $v$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In set theory, what does a singleton refer to? Back: A set with exactly one member. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set theory axiom is used to prove existence of singletons? Back: The pairing axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the pair set $\{u, v\}$ defined using set-builder notation? Back: $\{x \mid x = u \lor x = v\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Union Axiom ### Preliminary Form For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$ %%ANKI Basic What does the union axiom (preliminary form) state? Back: For any sets $a$ and $b$, there exists a set whose members are all in either $a$ or $b$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the union axiom (preliminary form) expressed using first-order logic? Back: $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$ Back: The union axiom (preliminary form). Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the union of sets $a$ and $b$ denoted? Back: $a \cup b$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What two set theory axioms prove existence of e.g. $\{x_1, x_2, x_3\}$? Back: The pairing axiom and union axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the union of set $a$ and $b$ defined using set-builder notation? Back: $\{x \mid x \in a \lor x \in b\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set operation is shaded green in the following venn diagram? ![[venn-diagram-union.png]] Back: $A \cup B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### General Form For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$ %%ANKI Basic What does the union axiom (general form) state? Back: For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the union axiom (general form) expressed using first-order logic? Back: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What advantage does the general form of the union axiom have over its prelimiary form? Back: The general form can handle infinite sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the preliminary form of the union axiom proven using the general form? Back: For any sets $a$ and $b$, $\bigcup \{a, b\} = a \cup b$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the result of $\bigcup \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$? Back: $\{2, 4, 6, 16, 26, 0\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the result of $\bigcup \varnothing$? Back: $\varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcup A$ represented in first-order logic? Back: $\{x \mid \exists b \in A, x \in b\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Roughly speaking, how does $\bigcup A$ adjust as $A$ gets larger? Back: $\bigcup A$ gets larger. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $A \subseteq B$, how do $\bigcup A$ and $\bigcup B$ relate? Back: $\bigcup A \subseteq \bigcup B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is the absolute complement of set $A$? Back: A class. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What kind of mathematical object is the relative complement of set $B$ in $A$? Back: A set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {1:Classes} are to {2:absolute} complements whereas {2:sets} are to {1:relative} complements. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What contradiction arises when arguing the absolute complement of set $A$ is a set? Back: The union of the complement with $A$ is the *class* of all sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Why is the absolute complement of sets rarely useful in set theory? Back: The absolute complement of a set isn't a set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set operation is shaded green in the following venn diagram? ![[venn-diagram-abs-comp.png]] Back: The absolute complement of $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Power Set Axiom For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$ %%ANKI Basic What does the power set axiom state? Back: For any set $a$, there exists a set whose members are exactly the subsets of $a$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the power set axiom expressed using first-order logic? Back: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$ Back: The power set axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $x \subseteq a$ rewritten using first-order logic and $\in$? Back: $\forall t, t \in x \Rightarrow t \in a$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the power set of set $a$ denoted? Back: $\mathscr{P}{a}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the power set of set $a$ defined using set-builder notation? Back: $\{x \mid x \subseteq a\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Subset Axioms For each formula $\_\_\_$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$ %%ANKI Basic What do the subset axioms state? Back: For each formula $\_\_\_$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $\_\_\_$ be a wff excluding $B$. How is its subset axiom stated in first-order logic? Back: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom(s)? $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$ Back: The subset axioms. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which axioms prove the existence of the union of two sets? Back: The union axiom. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which axioms prove the existence of the intersection of two sets? Back: The subset axioms. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the intersection of sets $A$ and $B$ denoted? Back: $A \cap B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the intersection of sets $a$ and $b$ defined using set-builder notation? Back: $\{x \mid x \in a \land x \in b\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which axioms prove the existence of the relative complement of two sets? Back: The subset axioms. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given sets $A$ and $B$, what does $A - B$ denote? Back: The relative complement of $B$ in $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the relative complement of set $B$ in $A$ denoted? Back: $A - B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is the relative complement of set $b$ in $a$ defined using set-builder notation? Back: $\{x \mid x \in a \land x \not\in b\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze Union is to the {union axiom} whereas intersection is to the {subset axioms}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The subset axioms ensure we do not construct what kind of mathematical object? Back: Classes. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\bigcap A$ represented in first-order logic? Back: $\{x \mid \forall b \in A, x \in b\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {1:$\forall$} is to {2:$\bigcap$} whereas {2:$\exists$} is to {1:$\bigcup$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the result of $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$? Back: $\{6\}$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does $\bigcap A$ adjust as $A$ gets larger? Back: $\bigcap A$ gets smaller. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic If $A \subseteq B$, how do $\bigcap A$ and $\bigcap B$ relate? Back: $\bigcap B \subseteq \bigcap A$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What class does $\bigcap \varnothing$ correspond to? Back: The class of all sets. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic *Why* does $\bigcap \varnothing$ present a problem? Back: Every set $x$ is a member of every member of $\varnothing$ (vacuously). Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze {$\bigcap \varnothing$} is to set theory as {division by zero} is to arithmetic. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set operation is shaded green in the following venn diagram? ![[venn-diagram-intersection.png]] Back: $A \cap B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What set operation is shaded green in the following venn diagram? ![[venn-diagram-rel-comp.png]] Back: $A - B$ 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). * “Russell’s Paradox,” in *Wikipedia*, April 18, 2024, [https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437](https://en.wikipedia.org/w/index.php?title=Russell%27s_paradox&oldid=1219576437). * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).