Algebra of sets and git flashcards.

c-declarations
Joshua Potter 2024-05-22 11:07:31 -06:00
parent b4e6daa052
commit 8f161f4f52
10 changed files with 234 additions and 6 deletions

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@ -129,7 +129,11 @@
"graph-subgraph.png", "graph-subgraph.png",
"graph-non-subgraph.png", "graph-non-subgraph.png",
"lcrs-nodes.png", "lcrs-nodes.png",
"binary-tree-nodes.png" "binary-tree-nodes.png",
"venn-diagram-union.png",
"venn-diagram-intersection.png",
"venn-diagram-rel-comp.png",
"venn-diagram-abs-comp.png"
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"Basic": [ "Basic": [

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- [ ] Sheet Music (10 min.) - [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem) - [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story) - [ ] Korean (Read 1 Story)
* Read "Chapter 8. The Trouble with Distributed Systems" in "Designing Data-Intensive Applications".
* Begin taking notes/creating flashcards on the [[algebra|algebra of sets]].
* Additional flashcards on git branching.

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@ -517,6 +517,30 @@ Reference: Scott Chacon, *Pro Git*, Second edition, The Experts Voice in Soft
<!--ID: 1709674569928--> <!--ID: 1709674569928-->
END%% END%%
%%ANKI
Basic
How many parents does an initial commit have?
Back: Zero.
Reference: Scott Chacon, *Pro Git*, Second edition, The Experts Voice in Software Development (New York, NY: Apress, 2014).
<!--ID: 1716397645567-->
END%%
%%ANKI
Basic
How many parents does a "normal" commit have?
Back: One.
Reference: Scott Chacon, *Pro Git*, Second edition, The Experts Voice in Software Development (New York, NY: Apress, 2014).
<!--ID: 1716397645568-->
END%%
%%ANKI
Basic
How many parents does a "merge" commit have?
Back: Two or more.
Reference: Scott Chacon, *Pro Git*, Second edition, The Experts Voice in Software Development (New York, NY: Apress, 2014).
<!--ID: 1716397645570-->
END%%
## Tags ## Tags
Tags are (possibly indirect) pointers to a git object. They *usually* point to a commit but aren't required to. There are two types of tags: Tags are (possibly indirect) pointers to a git object. They *usually* point to a commit but aren't required to. There are two types of tags:

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notes/set/algebra.md Normal file
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---
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).
<!--ID: 1716396060602-->
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%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

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@ -359,6 +359,15 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1715900348153--> <!--ID: 1715900348153-->
END%% 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%%
### General Form ### General Form
For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$ For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$
@ -435,6 +444,54 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716309007864--> <!--ID: 1716309007864-->
END%% 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 ## Power Set Axiom
For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$ For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
@ -656,6 +713,24 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716309007881--> <!--ID: 1716309007881-->
END%% 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%%
## Bibliography ## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).