More beta-reduction notes and set theory definitions.

c-declarations
Joshua Potter 2024-06-22 18:48:45 -06:00
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* Notes on [[beta-reduction#Normal Form|β-normal forms]]. * Notes on [[beta-reduction#Normal Form|β-normal forms]].
* Very basic notes on [[ars|abstract rewriting systems]]. * Very basic notes on [[ars|abstract rewriting systems]].
* Additional set theory definitions ([[functions#Restrictions|restrictions]] and [[functions#Images|images]]).

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%%ANKI %%ANKI
Basic Basic
How many contractions can make up a $\beta$-reduction? How many contractions make up a $\beta$-reduction?
Back: Zero or more. Back: Zero or more.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1719065602215--> <!--ID: 1719065602215-->
END%% END%%
%%ANKI
Basic
Is $x(\lambda u. uv)$ in $\beta$-normal form?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
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END%%
%%ANKI
Basic
Is $(\lambda u. uv)x$ in $\beta$-normal form?
Back: No.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
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END%%
%%ANKI
Basic
Why isn't $x(\lambda u. uv)$ in $\beta$-normal form?
Back: N/A. It is.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
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END%%
%%ANKI
Basic
Why isn't $(\lambda u. uv)x$ in $\beta$-normal form?
Back: Because $(\lambda u. uv)x$ is a $\beta$-redex.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
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END%%
## Bibliography ## Bibliography
* Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf). * Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).

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## Inverses ## Inverses
Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}.$$ Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$
%%ANKI %%ANKI
Basic Basic
What kind of mathematical object does the inverse operation apply to? What kind of mathematical object does the inverse operation apply to?
@ -617,6 +617,22 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1719017560113--> <!--ID: 1719017560113-->
END%% END%%
%%ANKI
Basic
How is set $\{\langle u, v \rangle \mid vAu\}$ more simply denoted?
Back: $A^{-1}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644290-->
END%%
%%ANKI
Basic
What does $\varnothing^{-1}$ evalute to?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
## Compositions ## Compositions
Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$ Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$
@ -676,6 +692,243 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1719017560123--> <!--ID: 1719017560123-->
END%% END%%
%%ANKI
Basic
How is set $\{\langle u, v \rangle \mid \exists t, uBt \land tAv \}$ more simply denoted?
Back: $A \circ B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644294-->
END%%
%%ANKI
Basic
Let $F$ be an arbitrary set. What is $F \circ \varnothing$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644295-->
END%%
%%ANKI
Basic
Let $F$ be an arbitrary set. What is $\varnothing \circ F$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
## Restrictions
Let $F$ and $A$ be arbitrary sets. The **restriction of $F$ to $A$** is the set $$F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$$
%%ANKI
Basic
What kind of mathematical object does the restriction operation apply to?
Back: Sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Cloze
$F \restriction A$ is the restriction of $F$ {to} $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644298-->
END%%
%%ANKI
Basic
What kind of mathematical object does the restriction operation emit?
Back: Relations.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
What is the "arity" of the restriction operation in set theory?
Back: $2$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is the restriction of $F$ to $A$ denoted?
Back: $F \restriction A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644301-->
END%%
%%ANKI
Basic
How is the restriction of $F$ to $A$ defined?
Back: $F \restriction A = \{\langle u, v \rangle \mid uFv \land u \in A\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644302-->
END%%
%%ANKI
Basic
Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction A)$?
Back: They are equal.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644303-->
END%%
%%ANKI
Basic
Consider function $F \colon A \rightarrow B$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction A)$?
Back: They are equal.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644304-->
END%%
%%ANKI
Basic
Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{dom}}F$ relate to $\mathop{\text{dom}}(F \restriction C)$?
Back: $\mathop{\text{dom}}(F \restriction C) \subseteq \mathop{\text{dom}}F$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644305-->
END%%
%%ANKI
Basic
How is $F \restriction A$ pronounced?
Back: The restriction of $F$ to $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644306-->
END%%
%%ANKI
Basic
Consider function $F \colon A \rightarrow B$ and set $C \subseteq A$. How does $\mathop{\text{ran}}F$ relate to $\mathop{\text{ran}}(F \restriction C)$?
Back: $\mathop{\text{ran}}(F \restriction C) \subseteq \mathop{\text{ran}}F$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644307-->
END%%
%%ANKI
Basic
How is set $\{\langle u, v \rangle \mid uAv \land u \in B\}$ more simply denoted?
Back: $A \restriction B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644308-->
END%%
%%ANKI
Basic
Let $F$ be an arbitrary set. What is $F \restriction \varnothing$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644309-->
END%%
## Images
Let $F$ and $A$ be sets. Then the **image of $F$ under $A$** is $$F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$$
%%ANKI
Basic
What kind of mathematical object does the image operation apply to?
Back: Sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644310-->
END%%
%%ANKI
Basic
What kind of mathematical object does the image operation emit?
Back: Sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644311-->
END%%
%%ANKI
Cloze
$F[\![A]\!]$ is the image of $F$ {under} $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644312-->
END%%
%%ANKI
Basic
What is the "arity" of the image operation in set theory?
Back: $2$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644313-->
END%%
%%ANKI
Basic
How is the image of $F$ under $A$ denoted?
Back: $F[\![A]\!]$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644314-->
END%%
%%ANKI
Basic
How is the image of $F$ under $A$ defined?
Back: $F[\![A]\!] = \{v \mid \exists u \in A, uFv\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644315-->
END%%
%%ANKI
Basic
How is the image of $F$ under $A$ defined in terms of restrictions?
Back: $F[\![A]\!] = \mathop{\text{ran}}(F \restriction A)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644316-->
END%%
%%ANKI
Basic
How is set $\{v \mid \exists u \in B, uAv\}$ more simply denoted?
Back: $A[\![B]\!]$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644317-->
END%%
%%ANKI
Basic
Enderton says "multiple-valued functions" are actually what?
Back: Relations.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644318-->
END%%
%%ANKI
Basic
Enderton says "multiple-valued functions" are actually what?
Back: Relations.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
END%%
%%ANKI
Basic
Enderton says "$F^{-1}(9) = \pm 3$" is preferably written in what way?
Back: $F^{-1}[\![\{9\}]\!] = \{-3, 3\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644319-->
END%%
%%ANKI
Basic
How is $F^{-1}[\![A]\!]$ defined using set-builder notation?
Back: $F^{-1}[\![A]\!] = \{x \in \mathop{\text{dom}} F \mid F(x) \in A \}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644320-->
END%%
%%ANKI
Basic
Let $F$ be an arbitrary set. What is $F[\![\varnothing]\!]$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1719103644321-->
END%%
## Bibliography ## Bibliography
* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163). * “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).