260 lines
8.0 KiB
Markdown
260 lines
8.0 KiB
Markdown
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---
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title: Axioms
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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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Enderton describes ten different axioms in total which serve as the foundation of our set theory.
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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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## 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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## 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 proves 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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## 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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## Bibliography
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* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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