More algebra of set identities and truth table analogs.

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notes/_journal/2024-02/2024-02-16.md

@@ -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.

notes/_journal/2024-05-27.md

@@ -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.

notes/_journal/2024-05/2024-05-22.md

@@ -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.

notes/algebra/sequences/delta-constant.md

@@ -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

notes/algebra/sequences/triangular-numbers.md

@@ -125,7 +125,7 @@ END%%
 
 %%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).
 <!--ID: 1709419325918-->

notes/algebra/set.md

@@ -115,6 +115,79 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--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).

notes/logic/equiv-trans.md

@@ -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.
 <!--ID: 1707316178714-->
 END%%

notes/logic/truth-tables.md

@@ -87,6 +87,38 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
 <!--ID: 1707311869003-->
 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.
+* 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).