notebook/notes/algebra/set.md

---
title: Algebra of Sets
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::set set
tags:
  - algebra
  - set
---

## Overview

The study of the operations of union ($\cup$), intersection ($\cap$), and set difference ($-$), together with the inclusion relation ($\subseteq$), goes by the **algebra of sets**.

%%ANKI
Basic
What three operators make up the algebra of sets?
Back: $\cup$, $\cap$, and $-$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%

%%ANKI
Basic
What *relation* is relevant in the algebra of sets?
Back: $\subseteq$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060605-->
END%%

## Laws

The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.

%%ANKI
Cloze
{$\cup$} is to algebra of sets whereas {$+$} is to algebra of real numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060607-->
END%%

%%ANKI
Cloze
{$\cap$} is to algebra of sets whereas {$\cdot$} is to algebra of real numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060609-->
END%%

%%ANKI
Cloze
{$-$} is to algebra of sets whereas {$-$} is to algebra of real numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060611-->
END%%

%%ANKI
Cloze
{$\subseteq$} is to algebra of sets whereas {$\leq$} is to algebra of real numbers.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060614-->
END%%

### Commutative Laws

For any sets $A$ and $B$, $$\begin{align*} A \cup B & = B \cup A \\ A \cap B & = B \cap A \end{align*}$$

%%ANKI
Basic
The commutative laws of the algebra of sets apply to what operators?
Back: $\cup$ and $\cap$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060616-->
END%%

%%ANKI
Basic
What does the union commutative law state?
Back: For any sets $A$ and $B$, $A \cup B = B \cup A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060618-->
END%%

%%ANKI
Basic
What does the intersection commutative law state?
Back: For any sets $A$ and $B$, $A \cap B = B \cap A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060620-->
END%%

### Associative Laws

For any sets $A$ and $B$, $$\begin{align*} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{align*}$$

%%ANKI
Basic
The associative laws of the algebra of sets apply to what operators?
Back: $\cup$ and $\cap$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060622-->
END%%

%%ANKI
Basic
What does the union associative law state?
Back: For any sets $A$, $B$, and $C$, $A \cup (B \cup C) = (A \cup B) \cup C$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060624-->
END%%

%%ANKI
Basic
What does the intersection associative law state?
Back: For any sets $A$, $B$, and $C$, $A \cap (B \cap C) = (A \cap B) \cap C$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716396060625-->
END%%

### Distributive Laws

For any sets $A$, $B$, and $C$, $$\begin{align*} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{align*}$$

%%ANKI
Basic
The distributive laws of the algebra of sets apply to what operators?
Back: $\cup$ and $\cap$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270441-->
END%%

%%ANKI
Cloze
The distributive law states {$A \cap (B \cup C)$} $=$ {$(A \cap B) \cup (A \cap C)$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270447-->
END%%

%%ANKI
Cloze
The distributive law states {$A \cup (B \cap C)$} $=$ {$(A \cup B) \cap (A \cup C)$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270452-->
END%%

### De Morgan's Laws

For any sets $A$, $B$, and $C$, $$\begin{align*} C - (A \cup B) & = (C - A) \cap (C - B) \\ C - (A \cap B) & = (C - A) \cup (C - B) \end{align*}$$

%%ANKI
Basic
The De Morgan's laws of the algebra of sets apply to what operators?
Back: $\cup$, $\cap$, and $-$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270457-->
END%%

%%ANKI
Cloze
De Morgan's law states that {$C - (A \cup B)$} $=$ {$(C - A) \cap (C - B)$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270461-->
END%%

%%ANKI
Cloze
De Morgan's law states that {$C - (A \cap B)$} $=$ {$(C - A) \cup (C - B)$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270466-->
END%%

%%ANKI
Cloze
For their respective De Morgan's laws, {$-$} is to the algebra of sets whereas {$\neg$} is to boolean algebra.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270473-->
END%%

%%ANKI
Cloze
For their respective De Morgan's laws, {$\cup$} is to the algebra of sets whereas {$\lor$} is to boolean algebra.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270480-->
END%%

%%ANKI
Cloze
For their respective De Morgan's laws, {$\cap$} is to the algebra of sets whereas {$\land$} is to boolean algebra.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803270485-->
END%%

## Bibliography

* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).