260 lines
8.0 KiB
Markdown
260 lines
8.0 KiB
Markdown
---
|
|
title: Axioms
|
|
TARGET DECK: Obsidian::STEM
|
|
FILE TAGS: set
|
|
tags:
|
|
- set
|
|
---
|
|
|
|
## Overview
|
|
|
|
Enderton describes ten different axioms in total which serve as the foundation of our set theory.
|
|
|
|
## Extensionality
|
|
|
|
If two sets have exactly the same members, then they are equal: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
|
|
%%ANKI
|
|
Basic
|
|
What does the extensionality axiom state?
|
|
Back: If two sets have exactly the same members, then they are equal.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649069247-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the extensionality axiom expressed using first-order logic?
|
|
Back: $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649734312-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following encodes which set theory axiom? $$\forall A, \forall B, (x \in A \Leftrightarrow x \in B) \Rightarrow A = B$$
|
|
Back: The extensionality axiom.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649069254-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How many sets exist with no members?
|
|
Back: Exactly one.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649069256-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which set theory axiom proves uniqueness of $\varnothing$?
|
|
Back: The extensionality axiom.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649069259-->
|
|
END%%
|
|
|
|
## Empty Set Axiom
|
|
|
|
There exists a set having no members: $$\exists B, \forall x, x \not\in B$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the empty set axiom state?
|
|
Back: There exists a set having no members.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649734322-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the empty set axiom expressed using first-order logic?
|
|
Back: $$\exists B, \forall x, x \not\in B$$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649734327-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
The following encodes which set theory axiom? $$\exists B, \forall x, x \not\in B$$
|
|
Back: The empty set axiom.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649734332-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
Which set theory axiom proves existence of $\varnothing$?
|
|
Back: The empty set axiom.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649069259-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
What two properties ensures definition $\varnothing$ is well-defined?
|
|
Back: The empty set exists and is unique.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715688034312-->
|
|
END%%
|
|
|
|
## Pairing Axiom
|
|
|
|
For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the pairing axiom state?
|
|
Back: For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--ID: 1715649734337-->
|
|
END%%
|
|
|
|
%%ANKI
|
|
Basic
|
|
How is the pairing axiom expressed using first-order logic?
|
|
Back: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--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%%
|
|
|
|
## Union Axiom
|
|
|
|
### Preliminary Form
|
|
|
|
For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both): $$\forall a, \forall b, \exists B, \forall x, (x \in B \Leftrightarrow x \in a \lor x \in b)$$
|
|
|
|
%%ANKI
|
|
Basic
|
|
What does the union axiom (preliminary form) state?
|
|
Back: For any sets $a$ and $b$, there exists a set whose members are all in either $a$ or $b$.
|
|
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
|
<!--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 proves 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%%
|
|
|
|
## 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%%
|
|
|
|
## Bibliography
|
|
|
|
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). |