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Index and function sets.

c-declarations
Joshua Potter 2024-07-12 05:29:41 -06:00
parent 65517ee3f8
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---
title: "2024-07-12"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Notes on [[set#Index Sets|index sets]] and [[set#Function Sets|function sets]].

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---
title: "2024-07-11"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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%%ANKI
Basic
Given `A[p:q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[p:r]` in terms of $n = q - p + 1$?
Back: $\lceil n / 2 \rceil$.
Back: $\lceil n / 2 \rceil$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467192-->
END%%
@ -299,7 +299,7 @@ END%%
%%ANKI
Basic
Given `A[p:q]` and $r = \lfloor (p + q) / 2 \rfloor$, what is the size of `A[r+1:q]` in terms of $n = q - p + 1$?
Back: $\lfloor n / 2 \rfloor$.
Back: $\lfloor n / 2 \rfloor$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467198-->
END%%

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@ -794,6 +794,304 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1718107987936-->
END%%
## Index Sets
Let $I$ be a set, called the **index set**. Let $F$ be a [[functions|function]] whose domain includes $I$. Then we define $$\bigcup_{i \in I} F(i) = \bigcup\,\{F(i) \mid i \in I\}$$
and, if $I \neq \varnothing$, $$\bigcap_{i \in I} F(i) = \bigcap\, \{F(i) \mid i \in I\}$$
%%ANKI
Basic
What name does $I$ go by in expression $\bigcup_{i \in I} F(i)$?
Back: The "index set".
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492681-->
END%%
%%ANKI
Basic
How is $\bigcup_{i \in I} F(i)$ alternatively denoted?
Back: $\bigcup_{i \in I} F_i$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492687-->
END%%
%%ANKI
Basic
What kind of mathematic object is $I$ in expression $\bigcup_{i \in I} F(i)$?
Back: A set.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492690-->
END%%
%%ANKI
Basic
What kind of mathematic object is $F$ in expression $\bigcup_{i \in I} F(i)$?
Back: A function.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492693-->
END%%
%%ANKI
Basic
How is $\bigcup_{i \in I} F_i$ alternatively denoted?
Back: $\bigcup_{i \in I} F(i)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782592276-->
END%%
%%ANKI
Basic
What kind of mathematic object is $F$ in expression $\bigcup_{i \in I} F_i$?
Back: A function.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782592281-->
END%%
%%ANKI
Basic
What is the domain of $F$ assumed to be in expression $\bigcup_{i \in I} F(i)$?
Back: Some superset of $I$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492696-->
END%%
%%ANKI
Basic
What condition must $I$ satisfy in expression $\bigcup_{i \in I} F(i)$?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492699-->
END%%
%%ANKI
Basic
Suppose $I = \{0, 1, 2\}$. What does $\bigcup_{i \in I} F(i)$ evaluate to?
Back: $F(0) \cup F(1) \cup F(2)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492702-->
END%%
%%ANKI
Basic
Suppose $I = \varnothing$. What does $\bigcup_{i \in I} F(i)$ evaluate to?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492705-->
END%%
%%ANKI
Basic
What kind of mathematic object is $F$ in expression $\bigcap_{i \in I} F(i)$?
Back: A function.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492709-->
END%%
%%ANKI
Basic
How is $\bigcap_{i \in I} F(i)$ often alternatively denoted?
Back: $\bigcap_{i \in I} F_i$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492712-->
END%%
%%ANKI
Basic
What is the domain of $F$ assumed to be in expression $\bigcap_{i \in I} F(i)$?
Back: Some superset of $I$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492716-->
END%%
%%ANKI
Basic
What condition must $I$ satisfy in expression $\bigcap_{i \in I} F(i)$?
Back: $I \neq \varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492720-->
END%%
%%ANKI
Basic
Suppose $I = \{0, 1, 2\}$. What does $\bigcap_{i \in I} F(i)$ evaluate to?
Back: $F(0) \cap F(1) \cap F(2)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492724-->
END%%
%%ANKI
Basic
Suppose $I = \varnothing$. What does $\bigcap_{i \in I} F(i)$ evaluate to?
Back: N/A. This is undefined.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782492727-->
END%%
%%ANKI
Basic
How is $\bigcap_{i \in I} F_i$ alternatively denoted?
Back: $\bigcap_{i \in I} F(i)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782592285-->
END%%
%%ANKI
Basic
What kind of mathematic object is $F$ in expression $\bigcap_{i \in I} F_i$?
Back: A function.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782592288-->
END%%
## Function Sets
For sets $A$ and $B$, the collection of functions $F$ from $A$ into $B$ is: $$^AB = \{F \mid F \colon A \rightarrow B\}$$
$^AB$ is read as "$B$-pre-$A$". It is often written as $B^A$ instead.
%%ANKI
Basic
For sets $A$ and $B$, how is set $B^A$ defined?
Back: $\{F \mid F \colon A \rightarrow B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782833225-->
END%%
%%ANKI
Basic
For sets $A$ and $B$, how is set $^AB$ defined?
Back: $\{F \mid F \colon A \rightarrow B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782923177-->
END%%
%%ANKI
Basic
For any function $F \colon A \rightarrow B$, $F$ is a subset of what other set?
Back: $A \times B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782833233-->
END%%
%%ANKI
Basic
For any function $F \colon A \rightarrow B$, $F$ is a member of what other set?
Back: $\mathscr{P}(A \times B)$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782833236-->
END%%
%%ANKI
Basic
For sets $A$ and $B$, how is set $B^A$ pronounced?
Back: As "$B$-pre-$A$".
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782923183-->
END%%
%%ANKI
Basic
Why prefer notation $B^A$ over $^AB$?
Back: The notation mirrors $|B|^{|A|}$, the number of elements in $B^A$ given both sets are finite.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607431-->
END%%
%%ANKI
Basic
For sets $A$ and $B$, how is set $^AB$ pronounced?
Back: As "$B$-pre-$A$".
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782923193-->
END%%
%%ANKI
Basic
Why prefer notation $^AB$ over $B^A$?
Back: Because the sets are written left-to-right, from domain to codomain.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607434-->
END%%
%%ANKI
Basic
*Why* is set $B^A$ denoted the way it is?
Back: If $A$ and $B$ are finite, then $B^A$ has $|B|^{|A|}$ members.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720782923188-->
END%%
%%ANKI
Basic
What is the domain of $^\omega\{0, 1\}$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607437-->
END%%
%%ANKI
Basic
What is the domain of a member of $^\omega\{0, 1\}$?
Back: $\omega$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607440-->
END%%
%%ANKI
Basic
What is the range of $\{0, 1\}^\omega$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607444-->
END%%
%%ANKI
Basic
What is the range of a member of $\{0, 1\}^\omega$?
Back: $\{0, 1\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607448-->
END%%
%%ANKI
Basic
What does $\varnothing^\varnothing$ evaluate to?
Back: $\{\varnothing\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607451-->
END%%
%%ANKI
Basic
For $A \neq \varnothing$, what does $\varnothing^A$ evaluate to?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607455-->
END%%
%%ANKI
Basic
For $A \neq \varnothing$, *why* does $\varnothing^A = \varnothing$?
Back: No function can map a nonempty domain to an empty range.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607459-->
END%%
%%ANKI
Basic
For $A \neq \varnothing$, what does $^\varnothing A$ evaluate to?
Back: $\{\varnothing\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607463-->
END%%
%%ANKI
Basic
For $A \neq \varnothing$, *why* does $^\varnothing A = \{\varnothing\}$?
Back: $\varnothing$ is the only function with empty domain.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1720783607468-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

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%%ANKI
Basic
What does a hash table collision refer to?
Back: Two keys hashing to the same slot.
Back: Two different keys hashing to the same slot.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180976-->
END%%

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%%ANKI
Basic
If a $\lambda$-term has $\beta$-normal forms $P$ and $Q$, what can be said about $P$ and $Q$?
If a $\lambda$-term has $\beta$-normal forms $P$ and $Q$, how do $P$ and $Q$ relate to one another?
Back: $P \equiv_\alpha Q$
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: 1719577152610-->