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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 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%%

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): 15965, 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): 15965, 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): 15965, 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): 15965, 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): 15965, 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) 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 A \in B, uFx is equivalently written as x \in {\{v \mid \exists A \in B, uFv\}}. 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: \{A \mid A \in 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: N/A. 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%%

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%%

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\}, \{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 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%%

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.

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%%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.

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%%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.

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%%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.

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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 Axiom of Choice. Reference: Herbert B. Enderton, Elements of Set Theory (New York: Academic Press, 1977).

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%%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).

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%%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).

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%%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).

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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).

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%%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).

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%%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).

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Bibliography