1150 lines
37 KiB
Markdown
1150 lines
37 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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Set theory begins with two primitive notions of sets and membership. Other axioms are defined relative to these concepts.
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%%ANKI
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Basic
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What are the two primitive notions of set theory?
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Back: Sets and membership.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717417781230-->
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END%%
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%%ANKI
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Basic
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How does Enderton describe a primitive notion?
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Back: An undefined concept other concepts are defined with.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717417781236-->
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END%%
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%%ANKI
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Basic
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Axioms can be thought of as doing what to primitive notions?
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Back: Divulging partial information about their meaning.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717417781239-->
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END%%
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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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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 $10$:
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* Roster notation: $\{2, 3, 5, 7\}$
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* Set-builder notation: $\{x \mid x < 10 \land x \text{ is prime}\}$
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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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%%ANKI
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Basic
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What does an atom refer to in set theory?
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Back: Any entity that is not a set but can exist in one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716494526269-->
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END%%
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%%ANKI
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Basic
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What German term is used alternatively for "atoms"?
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Back: Urelements.
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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).
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<!--ID: 1720998380904-->
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END%%
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%%ANKI
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Basic
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Can sets be members of urelements?
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Back: No.
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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).
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<!--ID: 1720998380907-->
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END%%
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%%ANKI
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Basic
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Can urelements be members of sets?
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Back: Yes.
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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).
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<!--ID: 1720998380909-->
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END%%
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%%ANKI
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Basic
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Can urelements be members of urelements?
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Back: No.
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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).
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<!--ID: 1720998380910-->
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END%%
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%%ANKI
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Basic
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Can sets be members of sets?
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Back: Yes.
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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).
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<!--ID: 1720998380911-->
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END%%
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%%ANKI
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Cloze
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An {atom} is to set theory as an {atomic} logical statement is to propositional logic.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716807316136-->
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END%%
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%%ANKI
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Cloze
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A {set} is to set theory as a {2:molecular} logical statement is to propositional logic.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716807316144-->
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END%%
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%%ANKI
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Basic
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What distinguishes a set from an atom?
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Back: An atom cannot contain other entitites.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716494526277-->
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END%%
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%%ANKI
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Basic
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What intuition is broken when a box is viewed as an atom?
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Back: When viewed as an atom, the box is no longer a container.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716494526280-->
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END%%
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%%ANKI
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Basic
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Enderton's exposition makes what assumption about the set of all atoms?
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Back: It is the empty set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1716494526284-->
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END%%
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%%ANKI
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Basic
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How are members of the following set defined using extensionality and first-order logic? $$B = \{P(x) \mid \phi(x)\}$$
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Back: $\forall x, P(x) \in B \Leftrightarrow \phi(x)$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624727-->
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END%%
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%%ANKI
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Basic
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How are members of the following set defined using extensionality and first-order logic? $$B = \{x \mid x < 5 \land x \text{ is prime}\}$$
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Back: $\forall x, x \in B \Leftrightarrow (x < 5 \land x \text{ is prime})$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624730-->
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END%%
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%%ANKI
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Cloze
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$P(x) = T$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624733-->
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END%%
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%%ANKI
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Cloze
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$\exists u \in A, uFx$ is equivalently written as $x \in$ {$F[\![A]\!]$}.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624735-->
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END%%
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%%ANKI
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Basic
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How is set $\{P(y) \mid y \in B\}$ interpreted?
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Back: As the set of $P(y)$ for all $y \in B$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624736-->
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END%%
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%%ANKI
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Basic
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Given function $P$, how is set $\{P(y) \mid y \in B\}$ more compactly denoted?
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Back: $P[\![B]\!]$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624737-->
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END%%
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%%ANKI
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Basic
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How is set $\{P(y) \mid \exists y \in B\}$ interpreted?
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Back: If $B$ is empty, the empty set. Otherwise as singleton $\{P(y)\}$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624738-->
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END%%
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%%ANKI
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Basic
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How many members are in set $\{P(y) \mid \exists y \in B\}$?
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Back: At most $1$.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720369624739-->
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END%%
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%%ANKI
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Basic
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In set-builder notation, the left side of $\{\ldots \mid \ldots\}$ denotes what?
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Back: The members of the set.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610010-->
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END%%
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%%ANKI
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Basic
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In set-builder notation, the right side of $\{\ldots \mid \ldots\}$ denotes what?
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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: 1720370610016-->
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END%%
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%%ANKI
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Basic
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How is set $\{v \mid \exists A \in B, v = A\}$ written more compactly?
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Back: $B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610022-->
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END%%
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%%ANKI
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Basic
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How is set $\{v \mid \exists A \in B, v \in A\}$ written more compactly?
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Back: $\bigcup B$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720370610028-->
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END%%
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%%ANKI
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Basic
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How is $\{A \mid A \in B\}$ rewritten with an existential in the entrance requirement?
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Back: $\{v \mid \exists A \in B \land v = A\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1720381621849-->
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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: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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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: $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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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? $$\forall A, \forall B, (\forall x, x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
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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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%%ANKI
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Basic
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What axiom is used to prove two sets are equal to one another?
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Back: Extensionality.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1717372494462-->
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END%%
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%%ANKI
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Basic
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Let $A$ and $B$ be sets. Proving the following is equivalent to showing what class is a set? $$\exists C, \forall y, (y \in C \Leftrightarrow y = \{x\} \times B \text{ for some } x \in A)$$
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Back: $\{\{x\} \times B \mid x \in A\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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<!--ID: 1718105051820-->
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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
|
||
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
|
||
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: Existence and uniqueness.
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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
|
||
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
|
||
Basic
|
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What does the pairing axiom state?
|
||
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
|
||
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).
|
||
<!--ID: 1715649734341-->
|
||
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).
|
||
<!--ID: 1715649734346-->
|
||
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).
|
||
<!--ID: 1715649734351-->
|
||
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).
|
||
<!--ID: 1715649734357-->
|
||
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).
|
||
<!--ID: 1715688034322-->
|
||
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).
|
||
<!--ID: 1715688034325-->
|
||
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).
|
||
<!--ID: 1715688034329-->
|
||
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).
|
||
<!--ID: 1715900348148-->
|
||
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).
|
||
<!--ID: 1715688034333-->
|
||
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).
|
||
<!--ID: 1715688034337-->
|
||
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).
|
||
<!--ID: 1715688034341-->
|
||
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).
|
||
<!--ID: 1715688034346-->
|
||
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).
|
||
<!--ID: 1715688034351-->
|
||
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).
|
||
<!--ID: 1715900348153-->
|
||
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).
|
||
<!--ID: 1716395245855-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does $\bigcup\,\{x\}$ evaluate to?
|
||
Back: $x$
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1723933434782-->
|
||
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).
|
||
<!--ID: 1716309007845-->
|
||
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).
|
||
<!--ID: 1716309007849-->
|
||
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).
|
||
<!--ID: 1716309007851-->
|
||
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).
|
||
<!--ID: 1716309007853-->
|
||
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).
|
||
<!--ID: 1716309007855-->
|
||
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).
|
||
<!--ID: 1716309007857-->
|
||
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).
|
||
<!--ID: 1716309007859-->
|
||
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).
|
||
<!--ID: 1716309007861-->
|
||
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).
|
||
<!--ID: 1716309007864-->
|
||
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).
|
||
<!--ID: 1716395245860-->
|
||
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).
|
||
<!--ID: 1716395245862-->
|
||
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).
|
||
<!--ID: 1716395245866-->
|
||
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).
|
||
<!--ID: 1716395245868-->
|
||
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).
|
||
<!--ID: 1716395245870-->
|
||
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).
|
||
<!--ID: 1716397645564-->
|
||
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).
|
||
<!--ID: 1715688034356-->
|
||
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).
|
||
<!--ID: 1715688034361-->
|
||
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).
|
||
<!--ID: 1715688034368-->
|
||
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).
|
||
<!--ID: 1715688034375-->
|
||
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).
|
||
<!--ID: 1715688034381-->
|
||
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).
|
||
<!--ID: 1715900348160-->
|
||
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).
|
||
<!--ID: 1716074312858-->
|
||
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).
|
||
<!--ID: 1716074312864-->
|
||
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).
|
||
<!--ID: 1716074312869-->
|
||
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).
|
||
<!--ID: 1716074312873-->
|
||
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).
|
||
<!--ID: 1716074312876-->
|
||
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).
|
||
<!--ID: 1716074312880-->
|
||
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).
|
||
<!--ID: 1716074312884-->
|
||
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).
|
||
<!--ID: 1716074312888-->
|
||
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).
|
||
<!--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%%
|
||
|
||
%%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).
|
||
<!--ID: 1716309007866-->
|
||
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).
|
||
<!--ID: 1716309007868-->
|
||
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).
|
||
<!--ID: 1716309007870-->
|
||
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).
|
||
<!--ID: 1716309007872-->
|
||
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).
|
||
<!--ID: 1716309007874-->
|
||
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).
|
||
<!--ID: 1716309007876-->
|
||
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).
|
||
<!--ID: 1716309007878-->
|
||
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).
|
||
<!--ID: 1716309007881-->
|
||
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).
|
||
<!--ID: 1716395245873-->
|
||
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).
|
||
<!--ID: 1716395245875-->
|
||
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).
|
||
<!--ID: 1717554445655-->
|
||
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).
|
||
<!--ID: 1717368558153-->
|
||
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).
|
||
<!--ID: 1717368558159-->
|
||
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).
|
||
<!--ID: 1717368558164-->
|
||
END%%
|
||
|
||
%%ANKI
|
||
Basic
|
||
What does $\bigcap\,\{x\}$ evaluate to?
|
||
Back: $x$
|
||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||
<!--ID: 1723925562044-->
|
||
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).
|
||
<!--ID: 1720964209614-->
|
||
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).
|
||
<!--ID: 1720964209620-->
|
||
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).
|
||
<!--ID: 1720964209624-->
|
||
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).
|
||
<!--ID: 1720964209627-->
|
||
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).
|
||
<!--ID: 1719681913526-->
|
||
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).
|
||
<!--ID: 1719681913527-->
|
||
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).
|
||
<!--ID: 1720964209631-->
|
||
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).
|
||
<!--ID: 1720964209636-->
|
||
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).
|
||
<!--ID: 1720964209640-->
|
||
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).
|
||
<!--ID: 1720964209644-->
|
||
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).
|
||
<!--ID: 1720964209648-->
|
||
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).
|
||
<!--ID: 1724486269578-->
|
||
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).
|
||
<!--ID: 1724486269581-->
|
||
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).
|
||
<!--ID: 1724486269585-->
|
||
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). |