--- title: Set TARGET DECK: Obsidian::STEM FILE TAGS: set tags: - set --- ## Overview Set theory begins with two primitive notions of sets and membership. Other axioms are defined relative to these concepts. %%ANKI Basic What are the two primitive notions of set theory? Back: Sets and membership. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does Enderton describe a primitive notion? Back: An undefined concept other concepts are defined with. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Axioms can be thought of as doing what to primitive notions? Back: Divulging partial information about their meaning. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How does Cormen et al. 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 Cormen et al. 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 Cormen et al. 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%% Sets are often denoted using **roster notation** in which members are specified explicitly in a comma-delimited list surrounded by curly braces. Alternatively, **abstraction** (or **set-builder notation**) defines sets using an **entrance requirement**. Examples of the set of prime numbers less than $10$: * Roster notation: $\{2, 3, 5, 7\}$ * Set-builder notation: $\{x \mid x < 10 \land x \text{ is prime}\}$ %%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 Basic What German term is used alternatively for "atoms"? Back: Urelements. Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009). END%% %%ANKI Basic Can sets be members of urelements? Back: No. Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009). END%% %%ANKI Basic Can urelements be members of sets? Back: Yes. Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009). END%% %%ANKI Basic Can urelements be members of urelements? Back: No. Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009). END%% %%ANKI Basic Can sets be members of sets? Back: Yes. Reference: Simon Hewitt, “A Cardinal Worry for Permissive Metaontology,” _Metaphysica_ 16, no. 2 (September 18, 2015): 159–65, [https://doi.org/10.1515/mp-2015-0009](https://doi.org/10.1515/mp-2015-0009). END%% %%ANKI Cloze An {atom} is to set theory as an {atomic} logical statement is to propositional logic. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze A {set} is to set theory as a {2:molecular} logical statement is to propositional logic. 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 the set of all atoms? Back: It is the empty set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How are members of the following set defined using extensionality and first-order logic? $$B = \{P(x) \mid \phi(x)\}$$ Back: $\forall x, P(x) \in B \Leftrightarrow \phi(x)$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How are members of the following set defined using extensionality and first-order logic? $$B = \{x \mid x < 5 \land x \text{ is prime}\}$$ Back: $\forall x, x \in B \Leftrightarrow (x < 5 \land x \text{ is prime})$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze $P(x) = T$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze $\exists u \in A, uFx$ is equivalently written as $x \in$ {$F[\![A]\!]$}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{P(y) \mid y \in B\}$ interpreted? Back: As the set of $P(y)$ for all $y \in B$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given function $P$, how is set $\{P(y) \mid y \in B\}$ more compactly denoted? Back: $P[\![B]\!]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{P(y) \mid \exists y \in B\}$ interpreted? Back: If $B$ is empty, the empty set. Otherwise as singleton $\{P(y)\}$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How many members are in set $\{P(y) \mid \exists y \in B\}$? Back: At most $1$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In set-builder notation, the left side of $\{\ldots \mid \ldots\}$ denotes what? Back: The members of the set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic In set-builder notation, the right side of $\{\ldots \mid \ldots\}$ denotes what? Back: The entrance requirement. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{v \mid \exists A \in B, v = A\}$ written more compactly? Back: $B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is set $\{v \mid \exists A \in B, v \in A\}$ written more compactly? Back: $\bigcup B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic How is $\{A \mid A \in B\}$ rewritten with an existential in the entrance requirement? Back: $\{v \mid \exists A \in B \land v = A\}$ 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: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ %%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: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The following encodes which set theory axiom? $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$ 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%% %%ANKI Basic What axiom is used to prove two sets are equal to one another? Back: Extensionality. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ and $B$ be sets. Proving the following is equivalent to showing what class is a set? $$\exists C, \forall y, (y \in C \Leftrightarrow y = \{x\} \times B \text{ for some } x \in A)$$ Back: $\{\{x\} \times B \mid x \in A\}$ 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: Existence and uniqueness. 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 $b$ (or both). 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%% %%ANKI Basic What does $\bigcup\,\{x\}$ evaluate to? Back: $x$ 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: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists a \in A, x \in a)$$ 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 preliminary 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: $\{x \mid \exists a \in A, x \in a\}$ 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%% %%ANKI Cloze Let $A$ be a set and $C = \bigcup\, \{ x \mid \_\_\_ \}$. Then $C$ {$\supseteq$} $A$ if $A$ satisfies the {entrance requirement}. 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%% %%ANKI Basic Let $A$ be a set. What does $\bigcup \mathscr{P} A$ evaluate to? Back: $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ be a set. *Why* does $\bigcup \mathscr{P} A = A$? Back: Because $\mathscr{P} A$ evaluates to the subsets of $A$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ be a set. What does $\bigcap \mathscr{P} A$ evaluate to? Back: $\varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Let $A$ be a set. *Why* does $\bigcap \mathscr{P} A = \varnothing$? Back: Because $\varnothing \in \mathscr{P} A$. 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 $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{6\}\}$? 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 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 Cloze Let $A$ be a set and $C = \bigcap\, \{ x \mid \_\_\_ \}$. Then $C$ {$\subseteq$} $A$ if $A$ satisfies the {entrance requirement}. 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%% %%ANKI Basic What set operation is shaded green in the following venn diagram? ![[venn-diagram-symm-diff.png]] Back: $A \mathop{\triangle} B$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic The "subset axioms" are more accurately classified as what? Back: An axiom schema. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is an axiom schema? Back: An infinite bundle of axioms. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Which of the set theory axioms are more accurately described as an axiom schema? Back: The subset axioms. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does $\bigcap\,\{x\}$ evaluate to? Back: $x$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Axiom of Choice This axiom assumes the existence of some choice function capable of selecting some element from a nonempty set. Note this axiom is controversial because it is non-constructive: there is no procedure we can follow to decide which element was chosen. %%ANKI Basic Why is the Axiom of Choice named the way it is? Back: It assumes the existence of some choice function. Reference: “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262). END%% %%ANKI Basic In Russell's analogy, why is AoC unnecessary to pick left shoes from an infinite set of shoe pairs? Back: The choice function can be defined directly, i.e. as "pick left shoe". Reference: “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262). END%% %%ANKI Basic In Russell's analogy, why is AoC necessary to pick socks from an infinite set of sock pairs? Back: There is no choice function to choose/prefer one sock from/over the other. Reference: “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262). END%% %%ANKI Basic What objects does Russell's analogy use when explaining AoC? Back: Pairs of shoes vs. pairs of (indistinguishable) socks. Reference: “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262). END%% ### Relation Form For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$. %%ANKI Basic What is AoC an acronym for? Back: The **A**xiom **o**f **C**hoice. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What does the Axiom of Choice (relation form) state? Back: For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic AoC (relation form) posits the existence of what mathematical object? Back: A function. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic Given relation $R$, AoC implies existence of function $F$. How does $F$ relate to $R$? Back: $F \subseteq R$ and $\mathop{\text{dom}} F = \mathop{\text{dom}} R$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ### Infinite Cartesian Product Form For any set $I$ and function $H$ with domain $I$, if $H(i) \neq \varnothing$ for all $i \in I$, then $\bigtimes_{i \in I} H(i) \neq \varnothing$. %%ANKI Basic What does the Axiom of Choice (infinite Cartesian product form) state? Back: For any set $I$ and function $H$ with domain $I$, if $H(i) \neq \varnothing$ for all $i \in I$, then $\bigtimes_{i \in I} H(i) \neq \varnothing$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the antecedent used in AoC (infinite Cartesian product form)? Back: $H(i) \neq \varnothing$ for all $i \in I$. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic What is the consequent used in AoC (infinite Cartesian product form)? Back: $\bigtimes_{i \in I} H(i) \neq \varnothing$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Infinity Axiom There exists an [[natural-numbers#Inductive Sets|inductive]] set: $$\exists A, [\varnothing \in A \land (\forall a \in A, a^+ \in A)]$$ %%ANKI Basic What does the infinity axiom state? Back: There exists an inductive set. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Cloze The {infinity} axiom asserts the existence of an {inductive set}. Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% %%ANKI Basic State the infinity axiom in FOL. Back: $\exists A, [\varnothing \in A \land (\forall a \in A, a^+ \in A)]$ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). END%% ## Bibliography * “Axiom of Choice,” in _Wikipedia_, July 8, 2024, [https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262](https://en.wikipedia.org/w/index.php?title=Axiom_of_choice&oldid=1233242262). * 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).