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---
title: Set
TARGET DECK: Obsidian::STEM
FILE TAGS: set
tags:
- set
---
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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
Basic
What are the two primitive notions of set theory?
Back: Sets and membership.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How does Enderton describe a primitive notion?
Back: An undefined concept other concepts are defined with.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Axioms can be thought of as doing what to primitive notions?
Back: Divulging partial information about their meaning.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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%%ANKI
Basic
How does Knuth define a *dynamic* set?
Back: As a set that can change over time.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Basic
How does Knuth distinguish mathematical sets from dynamic sets?
Back: The former is assumed to be unchanging.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Basic
How does Knuth define a dictionary?
Back: As a dynamic set that allows insertions, deletions, and membership tests.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Basic
Which of dynamic sets and dictionaries are more general?
Back: The dynamic set.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Basic
Is a dynamic set a dictionary?
Back: Not necessarily.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Basic
Is a dictionary a dynamic set?
Back: Yes.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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END%%
%%ANKI
Cloze
A dictionary supports {insertions}, {deletions}, and {membership testing}.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: adt::dynamic_set
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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 $10$:
* Roster notation: $\{2, 3, 5, 7\}$
* Set-builder notation: $\{x \mid x < 10 \land x \text { is prime } \}$
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%%ANKI
Basic
Define the set of prime numbers less than $10$ using abstraction.
Back: $\{x \mid x < 10 \land x \text { is prime } \}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Define the set of prime numbers less than $5$ using set-builder notation.
Back: $\{x \mid x < 5 \land x \text { is prime } \}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Define the set of prime numbers less than $5$ using roster notation.
Back: $\{2, 3\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
Define the set of prime numbers less than $5$ using abstraction.
Back: $\{x \mid x < 5 \land x \text { is prime } \}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What term describes the expression to the right of $\mid$ in set-builder notation?
Back: The entrance requirement.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What term refers to $\_\_\; x\; \_\_$ in $\{x \mid \_\_\; x\; \_\_\}$?
Back: The entrance requirement.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
The term "entrance requirement" refers to what kind of set notation?
Back: Set-builder/abstraction.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What name is given to set notation in which members are explicitly listed?
Back: Roster notation.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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%%ANKI
Basic
What does an atom refer to in set theory?
Back: Any entity that is not a set but can exist in one.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
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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END%%
%%ANKI
Cloze
A {set} is to set theory as a {2:molecular} logical statement is to propositional logic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What distinguishes a set from an atom?
Back: An atom cannot contain other entitites.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What intuition is broken when a box is viewed as an atom?
Back: When viewed as an atom, the box is no longer a container.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
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Enderton's exposition makes what assumption about the set of all atoms?
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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END%%
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%%ANKI
Basic
How are members of the following set defined using extensionality and first-order logic? $$B = \{P(x) \mid \phi(x)\}$$
Back: $\forall x, P(x) \in B \Leftrightarrow \phi(x)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How are members of the following set defined using extensionality and first-order logic? $$B = \{x \mid x < 5 \land x \text { is prime } \}$$
Back: $\forall x, x \in B \Leftrightarrow (x < 5 \land x \text { is prime })$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Cloze
$P(x)$ is equivalently written as $x \in$ {$\{v \mid P(v)\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Cloze
$\exists A \in B, uFx$ is equivalently written as $x \in$ {$\{v \mid \exists A \in B, uFv\}$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is set $\{P(y) \mid y \in B\}$ interpreted?
Back: As the set of $P(y)$ for all $y \in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How many members are in set $\{P(y) \mid y \in B\}$?
Back: As many as the number of unique $P(y)$ for each $y \in B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is set $\{P(y) \mid \exists y \in B\}$ interpreted?
Back: If $B$ is empty, the empty set. Otherwise as singleton $\{P(y)\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How many members are in set $\{P(y) \mid \exists y \in B\}$?
Back: At most $1$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
In set-builder notation, the left side of $\{\ldots \mid \ldots\}$ denotes what?
Back: The members of the set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
In set-builder notation, the right side of $\{\ldots \mid \ldots\}$ denotes what?
Back: The entrance requirement.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!-- ID: 1720370610016 -->
END%%
%%ANKI
Basic
How is set $\{v \mid \exists A \in B, v = A\}$ written more compactly?
Back: $\{A \mid A \in B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!-- ID: 1720370610022 -->
END%%
%%ANKI
Basic
How is set $\{v \mid \exists A \in B, v \in A\}$ written more compactly?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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
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%%
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%%ANKI
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%%
%%ANKI
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%%
%%ANKI
Basic
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How many sets exist with no members?
Back: Exactly one.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
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Which set theory axiom proves uniqueness of $\varnothing$?
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
Basic
What axiom is used to prove two sets are equal to one another?
Back: Extensionality.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!-- ID: 1717372494462 -->
END%%
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%%ANKI
Basic
Let $A$ and $B$ be sets. Proving the following is equivalent to showing what class is a set? $$\exists C, \forall y, (y \in C \Leftrightarrow y = \{x\} \times B \text{ for some } x \in A)$$
Back: $\{\{x\} \times B \mid x \in A\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
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## Empty Set Axiom
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
What does the empty set axiom state?
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%%
%%ANKI
Basic
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How is the empty set axiom expressed using first-order logic?
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%%
%%ANKI
Basic
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The following encodes which set theory axiom? $$\exists B, \forall x, x \not\in B$$
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%%
%%ANKI
Basic
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Which set theory axiom proves existence of $\varnothing$?
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%%
%%ANKI
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%%
%%ANKI
Basic
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How is the empty set defined using set-builder notation?
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
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%%
%%ANKI
Basic
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How is the pairing axiom expressed using first-order logic?
Back: $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
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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%%
%%ANKI
Basic
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The following encodes which set theory axiom? $$\forall u, \forall v, \exists B, \forall x, (x \in B \Leftrightarrow x = u \lor x = v)$$
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.
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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%%
%%ANKI
Basic
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What set theory axiom is used to prove existence of singletons?
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%%
%%ANKI
Basic
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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).
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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?
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Back: For any sets $a$ and $b$, there exists a set whose members are all in either $a$ or $b$ (or both).
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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%%
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%%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%%
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### 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?
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Back: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists a \in A, x \in a)$$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!-- ID: 1716309007849 -->
END%%
%%ANKI
Basic
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What advantage does the general form of the union axiom have over its preliminary form?
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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?
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Back: $\{x \mid \exists a \in A, x \in a\}$
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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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).
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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%%
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%%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).
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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%%
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## 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?
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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).
<!-- 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.
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Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
2024-05-19 21:07:38 +00:00
<!-- ID: 1716074312909 -->
2024-05-17 18:33:05 +00:00
END%%
2024-05-21 18:40:04 +00:00
%%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
2024-06-06 13:14:13 +00:00
What is the result of $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{6\}\}$?
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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?
2024-06-03 13:55:29 +00:00
Back: Every set is a member of every member of $\varnothing$ (vacuously).
2024-05-21 18:40:04 +00:00
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%%
2024-05-22 17:07:31 +00:00
%%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%%
2024-06-06 13:14:13 +00:00
%%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%%
2024-06-03 13:55:29 +00:00
%%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%%
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## Axiom of Choice
### 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%%
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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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* “Russell’ *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 ).
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
