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Unions/intersections and distributivity of disjunction.

c-declarations
Joshua Potter 2024-05-21 12:40:04 -06:00
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notes/_journal/2024-05-21.md
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@ -8,6 +8,7 @@ title: "2024-05-21"
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
* TODO: Hash tables
* TODO: Arbitrary unions and intersections
* TODO: Distributitivity of Disjunction
* High-level notes/flashcards on [[hashing/index|hash tables]].
* Flashcards on arbitrary unions and intersections.
* Notes on [[pred-trans#Distributivity of Disjunction|Distributivity of Disjunction]].
* Watched [Lecture #09 - Concurrent Indexes](https://www.youtube.com/watch?v=5KClozM1jjw) on databases.

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notes/hashing/direct-addressing.md
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@ -10,14 +10,6 @@ tags:
Given a universe of keys $U = \{0, 1, \ldots, m - 1\}$, a **direct-address table** has $m$ **slots**. Each slot corresponds to a key in universe $U$.
%%ANKI
Basic
With respect to hashing, what does the "universe" of keys refer to?
Back: Every potential key that may be inserted into the underlying dictionary.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716046153757-->
END%%
%%ANKI
Basic
Given universe $U$, how many slots must a direct-address table have?
@ -63,6 +55,14 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1716046153781-->
END%%
%%ANKI
Basic
What is the worst-cast runtime complexity of direct-address table searches?
Back: $O(1)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180982-->
END%%
%%ANKI
Basic
Write pseudocode to insert $x$ into direct-address table `T[0:m-1]`.
@ -76,6 +76,14 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1716046153785-->
END%%
%%ANKI
Basic
What is the worst-case runtime complexity of direct-address table insertions?
Back: $O(1)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180983-->
END%%
%%ANKI
Basic
Write pseudocode to delete $x$ from direct-address table `T[0:m-1]`.
@ -89,6 +97,30 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1716046153789-->
END%%
%%ANKI
Basic
What is the worst-cast runtime complexity of direct-address table deletions?
Back: $O(1)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180984-->
END%%
%%ANKI
Basic
In what situation does direct addressing waste space?
Back: When the number of keys used is much less than the size of the universe.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180986-->
END%%
%%ANKI
Basic
In what situation is direct addressing impossible?
Back: When the size of the universe is too large to hold in memory.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180987-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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notes/hashing/index.md
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@ -1,5 +1,142 @@
---
title: Hashing
TARGET DECK: Obsidian::STEM
FILE TAGS: hashing
tags:
- hash
- hashing
---
## Overview
A **hash table** `T[0:m-1]` uses a **hash function** to map a universe of keys into slots of the hash table. It can be seen as a generalization of direct addressing (which has "hash function" $h(k) = k$).
%%ANKI
Basic
With respect to hashing, what does the "universe" of keys refer to?
Back: Every potential key that may be inserted into the underlying dictionary.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716046153757-->
END%%
%%ANKI
Basic
What name is given to each position in a hash table?
Back: A slot.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180959-->
END%%
%%ANKI
Basic
Given a hash table with hash function $h$, the element at slot $k$ has what key?
Back: A key $k'$ such that $h(k') = k$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180961-->
END%%
%%ANKI
Basic
Given a hash table with hash function $h$, an element with key $k$ is placed in what slot?
Back: $h(k)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180962-->
END%%
%%ANKI
Basic
Given a hash table `T[0:m-1]`, what is the domain of a hash function?
Back: The universe of keys.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180964-->
END%%
%%ANKI
Basic
Given a hash table `T[0:m-1]`, what is the codomain of a hash function?
Back: $\{0, \ldots, m - 1\}$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180965-->
END%%
%%ANKI
Basic
What does a hash value refer to?
Back: The result produced by a hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180967-->
END%%
%%ANKI
Basic
What distinguishes a slot from a hash value?
Back: The former is a memory address. The latter is the result of a hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180968-->
END%%
%%ANKI
Basic
What relationship exists between slots and hash values?
Back: A slot is often referred to by a hash value.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180970-->
END%%
%%ANKI
Cloze
Given hash function $h$, key $k$ {hashes} to slot $h(k)$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180971-->
END%%
%%ANKI
Basic
What combinatorial concept is used to prove the presence of hash table collisions?
Back: The pigeonhole principle.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180973-->
END%%
%%ANKI
Basic
When *must* there exist hash table collisions?
Back: When the number of hashed keys is greater than the number of slots.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180974-->
END%%
%%ANKI
Basic
What does a hash table collision refer to?
Back: Two 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%%
%%ANKI
Basic
With respect to hash tables, what imagery is invoked by the term "hash"?
Back: Random mixing and chopping.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180977-->
END%%
%%ANKI
Basic
Are hash tables or direct-address tables more general?
Back: Hash tables.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180979-->
END%%
%%ANKI
Basic
How is a direct-address table reinterpreted as a hash table?
Back: It's a hash table with hash function $h(k) = k$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1716307180980-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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notes/programming/pred-trans.md
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@ -313,14 +313,14 @@ Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q \land R) = wp(S, Q) \lan
Basic
What does Distributivity of Conjunction state?
Back: Given command $S$ and predicates $Q$ and $R$, $wp(S, Q \land R) = wp(S, Q) \land wp(S, R)$.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047060-->
END%%
%%ANKI
Cloze
Distributivity of Conjunction states {$wp(S, Q \land R)$} $=$ {$wp(S, Q) \land wp(S, R)$}.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047062-->
END%%
@ -328,7 +328,7 @@ END%%
Basic
In Gries's exposition, is Distributivity of Conjunction taken as an axiom or a theorem?
Back: An axiom.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047064-->
END%%
@ -336,7 +336,7 @@ END%%
Basic
Is $wp(S, Q) \land wp(S, R) \Rightarrow wp(S, Q \land R)$ true if $S$ is nondeterministic?
Back: Yes.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047065-->
END%%
@ -344,18 +344,10 @@ END%%
Basic
Is $wp(S, Q \land R) \Rightarrow wp(S, Q) \land wp(S, R)$ true if $S$ is nondeterministic?
Back: Yes.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047067-->
END%%
%%ANKI
Basic
What does it mean for command $S$ to be nondeterministic?
Back: Execution may not be the same even if begun in the same state.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047068-->
END%%
### Law of Monotonicity
Given command $S$ and predicates $Q$ and $R$, if $Q \Rightarrow R$, then $wp(S, Q) \Rightarrow wp(S, R)$.
@ -389,6 +381,129 @@ Reference: Reference: Gries, David. *The Science of Programming*. Texts and Mon
<!--ID: 1716227332870-->
END%%
### Distributivity of Disjunction
Given command $S$ and predicates $Q$ and $R$, $$wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$$
%%ANKI
Basic
What does Distributivity of Disjunction state?
Back: Given command $S$ and predicates $Q$ and $R$, $wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927694-->
END%%
%%ANKI
Cloze
Distributivity of Disjunction states {1:$wp(S, Q) \lor wp(S, r)$} $\Rightarrow$ {1:$wp(S, Q \lor R)$}.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927697-->
END%%
%%ANKI
Basic
In Gries's exposition, is Distributivity of Disjunction taken as an axiom or a theorem?
Back: A theorem.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927698-->
END%%
%%ANKI
Basic
Is $wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$ true if $S$ is nondeterministic?
Back: No.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927700-->
END%%
%%ANKI
Basic
Is $wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$ true if $S$ is nondeterministic?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927701-->
END%%
%%ANKI
Basic
Is $wp(S, Q \lor R) \Rightarrow wp(S, Q) \lor wp(S, R)$ true if $S$ is deterministic?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927703-->
END%%
%%ANKI
Basic
Is $wp(S, Q) \lor wp(S, R) \Rightarrow wp(S, Q \lor R)$ true if $S$ is deterministic?
Back: Yes.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927710-->
END%%
%%ANKI
Basic
What command does Gries use to demonstrate nondeterminism?
Back: The flipping of a coin.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927712-->
END%%
%%ANKI
Basic
What does it mean for command $S$ to be nondeterministic?
Back: Execution may not be the same even if begun in the same state.
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1715969047068-->
END%%
%%ANKI
Basic
Let $S$ flip a coin and $Q$ be flipping heads. What is $wp(S, Q)$?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927713-->
END%%
%%ANKI
Basic
Let $S$ flip a coin and $Q$ be flipping tails. What is $wp(S, Q)$?
Back: $F$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927715-->
END%%
%%ANKI
Basic
Let $S$ flip a coin, $Q$ be flipping heads, and $R$ be flipping tails. What is $wp(S, Q \lor R)$?
Back: $T$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927716-->
END%%
%%ANKI
Basic
What determines the direction of implication in Distributivity of Disjunction?
Back: $F \Rightarrow T$ evaluates truthily but $T \Rightarrow F$ does not.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716310927718-->
END%%
%%ANKI
Basic
*Why* does Distributivity of Disjunction use an implication instead of equality?
Back: Because the underlying command may be nondeterministic.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716311034191-->
END%%
%%ANKI
Basic
*When* does Distributivity of Disjunction hold under equality (instead of implication)?
Back: When the underlying command is deterministic.
Reference: Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1716311034194-->
END%%
## Bibliography
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

8
notes/set/classes.md
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@ -138,6 +138,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716237736487-->
END%%
%%ANKI
Basic
Why is "$x$ is an int definable in one line of type" an invalid entrance requirement?
Back: Because this sentence cannot be expressed in predicate logic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007883-->
END%%
## Russell's Paradox
Let $R = \{x \mid x \not\in x\}$. Then $R \in R \Leftrightarrow R \not\in R$.

138
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@ -359,6 +359,82 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1715900348153-->
END%%
### General Form
For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$
%%ANKI
Basic
What does the union axiom (general form) state?
Back: For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007845-->
END%%
%%ANKI
Basic
How is the union axiom (general form) expressed using first-order logic?
Back: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007849-->
END%%
%%ANKI
Basic
What advantage does the general form of the union axiom have over its prelimiary form?
Back: The general form can handle infinite sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007851-->
END%%
%%ANKI
Basic
How is the preliminary form of the union axiom proven using the general form?
Back: For any sets $a$ and $b$, $\bigcup \{a, b\} = a \cup b$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007853-->
END%%
%%ANKI
Basic
What is the result of $\bigcup \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$?
Back: $\{2, 4, 6, 16, 26, 0\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007855-->
END%%
%%ANKI
Basic
What is the result of $\bigcup \varnothing$?
Back: $\varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007857-->
END%%
%%ANKI
Basic
How is $\bigcup A$ represented in first-order logic?
Back: $\{x \mid \exists b \in A, x \in b\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007859-->
END%%
%%ANKI
Basic
Roughly speaking, how does $\bigcup A$ adjust as $A$ gets larger?
Back: $\bigcup A$ gets larger.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007861-->
END%%
%%ANKI
Basic
If $A \subseteq B$, how do $\bigcup A$ and $\bigcup B$ relate?
Back: $\bigcup A \subseteq \bigcup B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007864-->
END%%
## Power Set Axiom
For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\forall a, \exists B, \forall x, (x \in B \Leftrightarrow x \subseteq a)$$
@ -518,6 +594,68 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716074312909-->
END%%
%%ANKI
Basic
How is $\bigcap A$ represented in first-order logic?
Back: $\{x \mid \forall b \in A, x \in b\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007866-->
END%%
%%ANKI
Cloze
{1:$\forall$} is to {2:$\bigcap$} whereas {2:$\exists$} is to {1:$\bigcup$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007868-->
END%%
%%ANKI
Basic
What is the result of $\bigcap \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$?
Back: $\{6\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007870-->
END%%
%%ANKI
Basic
How does $\bigcap A$ adjust as $A$ gets larger?
Back: $\bigcap A$ gets smaller.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007872-->
END%%
%%ANKI
Basic
If $A \subseteq B$, how do $\bigcap A$ and $\bigcap B$ relate?
Back: $\bigcap B \subseteq \bigcap A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1716309007874-->
END%%
%%ANKI
Basic
What class does $\bigcap \varnothing$ correspond to?
Back: The class of all sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Basic
*Why* does $\bigcap \varnothing$ present a problem?
Back: Every set $x$ is a member of every member of $\varnothing$ (vacuously).
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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Cloze
{$\bigcap \varnothing$} is to set theory as {division by zero} is to arithmetic.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).