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Set Obsidian::STEM set
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%%

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%%

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%%

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%%

Bibliography