More algebra of set identities and truth table analogs.

c-declarations
Joshua Potter 2024-05-27 03:58:08 -06:00
parent e87168b297
commit 47519891a4
11 changed files with 135 additions and 15 deletions

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"Basic": [

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@ -12,4 +12,4 @@ title: "2024-02-16"
* Read "Queries, Modeling, and Transformation" of "Fundamentals of Data Engineering".
* Deleted many notes on shifting. Will wait until I look deeper into the actual representations of integral types before re-introducing.
* Spent time deriving properties of floor/ceiling functions and analogues in C. Added many more notes on partitioning arrays in halves.
* Spent time deriving properties of floor/ceiling functions and analogs in C. Added many more notes on partitioning arrays in halves.

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@ -0,0 +1,11 @@
---
title: "2024-05-27"
---
- [ ] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
* More algebra of sets identities and analogs between membership tables and truth tables.

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@ -9,6 +9,6 @@ title: "2024-05-22"
- [ ] 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]].
* Begin taking notes/creating flashcards on the [[set|algebra of sets]].
* Additional flashcards on git branching.
* Slight progress on Hide and Seek flutter app.

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@ -18,7 +18,7 @@ A sequence is said to be **$\Delta^k$-constant** if the $k$th differences are co
> The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $\Delta^k$-constant.
This is the discrete analogue to (continuous) derivatives of polynomials.
This is the discrete analog to (continuous) derivatives of polynomials.
%%ANKI
Basic
@ -77,7 +77,7 @@ END%%
%%ANKI
Basic
$\Delta^k$-constant sequences are a discrete analogue to what calculus concept?
$\Delta^k$-constant sequences are a discrete analog to what calculus concept?
Back: Derivatives.
Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
Tags: calculus

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%%ANKI
Basic
What polygonal sequence is the summation analogue of factorial?
What polygonal sequence is the summation analog of factorial?
Back: The triangular numbers.
Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122](https://en.wikipedia.org/w/index.php?title=Triangular_number&oldid=1195279122).
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@ -115,6 +115,79 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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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).
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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).
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END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

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@ -367,7 +367,7 @@ END%%
%%ANKI
Basic
How is e.g. the Law of Implication proven in the system of evaluation?
Back: With truth tables
Back: With truth tables.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%

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@ -87,6 +87,38 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
%%ANKI
Basic
What analog to truth tables is found in the algebra of sets?
Back: Membership tables.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803633023-->
END%%
%%ANKI
Cloze
{Truth} tables are to propositions whereas {membership} tables are to set identities.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803633029-->
END%%
%%ANKI
Basic
How many rows are in the truth table of identity $\neg (a \Rightarrow b) \Leftrightarrow c$?
Back: $2^3 = 8$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716803798112-->
END%%
%%ANKI
Basic
How many rows are in the membership table of identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$?
Back: $2^3 = 8$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716803798123-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).