525 lines
16 KiB
Markdown
525 lines
16 KiB
Markdown
---
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title: Set
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TARGET DECK: Obsidian::STEM
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FILE TAGS: set
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tags:
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- set
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---
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## Overview
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%%ANKI
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Basic
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How does Knuth define a *dynamic* set?
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Back: As a set that can change over time.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070055-->
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END%%
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%%ANKI
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Basic
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How does Knuth distinguish mathematical sets from dynamic sets?
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Back: The former is assumed to be unchanging.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070059-->
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END%%
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%%ANKI
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Basic
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How does Knuth define a dictionary?
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Back: As a dynamic set that allows insertions, deletions, and membership tests.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070063-->
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END%%
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%%ANKI
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Basic
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Which of dynamic sets and dictionaries are more general?
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Back: The dynamic set.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070067-->
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END%%
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%%ANKI
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Basic
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Is a dynamic set a dictionary?
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Back: Not necessarily.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070071-->
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END%%
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%%ANKI
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Basic
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Is a dictionary a dynamic set?
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Back: Yes.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070077-->
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END%%
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%%ANKI
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Cloze
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A dictionary supports {insertions}, {deletions}, and {membership testing}.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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Tags: adt::dynamic_set
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<!--ID: 1715432070083-->
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END%%
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%%ANKI
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Basic
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Define the set of prime numbers less than $10$ using abstraction.
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Back: $\{x \mid x < 10 \land x \text{ is prime}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028616-->
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END%%
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%%ANKI
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Basic
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Define the set of prime numbers less than $5$ using set-builder notation.
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Back: $\{x \mid x < 5 \land x \text{ is prime}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028645-->
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END%%
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%%ANKI
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Basic
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Define the set of prime numbers less than $5$ using roster notation.
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Back: $\{2, 3\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028649-->
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END%%
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%%ANKI
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Basic
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Define the set of prime numbers less than $5$ using abstraction.
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Back: $\{x \mid x < 5 \land x \text{ is prime}\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028652-->
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END%%
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%%ANKI
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Basic
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What term describes the expression to the right of $\mid$ in set-builder notation?
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Back: The entrance requirement.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028656-->
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END%%
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%%ANKI
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Basic
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What term refers to $\_\_\; x\; \_\_$ in $\{x \mid \_\_\; x\; \_\_\}$?
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Back: The entrance requirement.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028659-->
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END%%
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%%ANKI
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Basic
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The term "entrance requirement" refers to what kind of set notation?
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Back: Set-builder/abstraction.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028663-->
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END%%
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%%ANKI
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Basic
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What name is given to set notation in which members are explicitly listed?
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Back: Roster notation.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715786028667-->
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END%%
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## Extensionality
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If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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%%ANKI
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Basic
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What does the extensionality axiom state?
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Back: If two sets have exactly the same members, then they are equal.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649069247-->
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END%%
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%%ANKI
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Basic
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How is the extensionality axiom expressed using first-order logic?
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Back: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734312-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom? $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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Back: The extensionality axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649069254-->
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END%%
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%%ANKI
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Basic
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How many sets exist with no members?
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Back: Exactly one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649069256-->
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END%%
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%%ANKI
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Basic
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Which set theory axiom proves uniqueness of $\varnothing$?
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Back: The extensionality axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649069259-->
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END%%
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## Empty Set Axiom
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There exists a set having no members: $$\exists B, \forall x, x \not\in B$$
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%%ANKI
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Basic
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What does the empty set axiom state?
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Back: There exists a set having no members.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734322-->
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END%%
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%%ANKI
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Basic
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How is the empty set axiom expressed using first-order logic?
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Back: $$\exists B, \forall x, x \not\in B$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734327-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom? $$\exists B, \forall x, x \not\in B$$
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Back: The empty set axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734332-->
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END%%
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%%ANKI
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Basic
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Which set theory axiom proves existence of $\varnothing$?
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Back: The empty set axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649069259-->
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END%%
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%%ANKI
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Basic
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What two properties ensures definition $\varnothing$ is well-defined?
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Back: The empty set exists and is unique.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034312-->
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END%%
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%%ANKI
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Basic
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How is the empty set defined using set-builder notation?
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Back: $\{x \mid x \neq x\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715900348141-->
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END%%
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## Pairing Axiom
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For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
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%%ANKI
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Basic
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What does the pairing axiom state?
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Back: For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734337-->
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END%%
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%%ANKI
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Basic
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How is the pairing axiom expressed using first-order logic?
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Back: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734341-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom? $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
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Back: The pairing axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734346-->
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END%%
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%%ANKI
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Basic
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Which set theory axiom proves existence of set $\{x, y\}$ where $x \neq y$?
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Back: The pairing axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734351-->
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END%%
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%%ANKI
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Basic
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Which set theory axiom proves existence of set $\{x\}$?
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Back: The pairing axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715649734357-->
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END%%
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%%ANKI
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Basic
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For sets $u$ and $v$, what name is given to set $\{u, v\}$?
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Back: The pair set of $u$ and $v$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034322-->
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END%%
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%%ANKI
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Basic
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In set theory, what does a singleton refer to?
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Back: A set with exactly one member.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034325-->
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END%%
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%%ANKI
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Basic
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What set theory axiom is used to prove existence of singletons?
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Back: The pairing axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034329-->
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END%%
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%%ANKI
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Basic
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How is the pair set $\{u, v\}$ defined using set-builder notation?
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Back: $\{x \mid x = u \lor x = v\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715900348148-->
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END%%
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## Union Axiom
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### Preliminary Form
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For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
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%%ANKI
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Basic
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What does the union axiom (preliminary form) state?
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Back: For any sets $a$ and $b$, there exists a set whose members are all in either $a$ or $b$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034333-->
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END%%
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%%ANKI
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Basic
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How is the union axiom (preliminary form) expressed using first-order logic?
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Back: $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034337-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom? $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
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Back: The union axiom (preliminary form).
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034341-->
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END%%
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%%ANKI
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Basic
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How is the union of sets $a$ and $b$ denoted?
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Back: $a \cup b$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034346-->
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END%%
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%%ANKI
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Basic
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What two set theory axioms prove existence of e.g. $\{x_1, x_2, x_3\}$?
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Back: The pairing axiom and union axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034351-->
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END%%
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%%ANKI
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Basic
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How is the union of set $a$ and $b$ defined using set-builder notation?
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Back: $\{x \mid x \in a \lor x \in b\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715900348153-->
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END%%
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## Power Set Axiom
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For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
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%%ANKI
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Basic
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What does the power set axiom state?
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Back: For any set $a$, there exists a set whose members are exactly the subsets of $a$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034356-->
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END%%
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%%ANKI
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Basic
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How is the power set axiom expressed using first-order logic?
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Back: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034361-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom? $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
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Back: The power set axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034368-->
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END%%
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%%ANKI
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Basic
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How is $x \subseteq a$ rewritten using first-order logic and $\in$?
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Back: $\forall t, t \in x \Rightarrow t \in a$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034375-->
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END%%
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%%ANKI
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Basic
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How is the power set of set $a$ denoted?
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Back: $\mathscr{P}{a}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715688034381-->
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END%%
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%%ANKI
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Basic
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How is the power set of set $a$ defined using set-builder notation?
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Back: $\{x \mid x \subseteq a\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1715900348160-->
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END%%
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## Subset Axioms
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For each formula $\_\_\_$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$
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%%ANKI
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Basic
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What do the subset axioms state?
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Back: For each formula $\_\_\_$ not containing $B$, the following is an axiom: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312858-->
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END%%
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%%ANKI
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Basic
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Let $\_\_\_$ be a wff excluding $B$. How is its subset axiom stated in first-order logic?
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Back: $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312864-->
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END%%
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%%ANKI
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Basic
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The following encodes which set theory axiom(s)? $$\forall t_1, \cdots, \forall t_k, \forall c, \exists B, \forall x, (x \in B \Leftrightarrow x \in c \land \_\_\_)$$
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Back: The subset axioms.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312869-->
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END%%
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%%ANKI
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Basic
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Which axioms prove the existence of the union of two sets?
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Back: The union axiom.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312873-->
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END%%
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%%ANKI
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Basic
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Which axioms prove the existence of the intersection of two sets?
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Back: The subset axioms.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312876-->
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END%%
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%%ANKI
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Basic
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How is the intersection of sets $A$ and $B$ denoted?
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Back: $A \cap B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312880-->
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END%%
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%%ANKI
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Basic
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How is the intersection of sets $a$ and $b$ defined using set-builder notation?
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Back: $\{x \mid x \in a \land x \in b\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312884-->
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END%%
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%%ANKI
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Basic
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Which axioms prove the existence of the relative complement of two sets?
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Back: The subset axioms.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716074312888-->
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END%%
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%%ANKI
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Basic
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Given sets $A$ and $B$, what does $A - B$ denote?
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Back: The relative complement of $B$ in $A$.
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1716074312893-->
|
||
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).
|
||
<!--ID: 1716074312897-->
|
||
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).
|
||
<!--ID: 1716074312901-->
|
||
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).
|
||
<!--ID: 1716074312905-->
|
||
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).
|
||
<!--ID: 1716074312909-->
|
||
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). |