diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 7e41029..4e30c18 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -309,7 +309,7 @@ "_journal/2024-03-18.md": "8479f07f63136a4e16c9cd07dbf2f27f", "_journal/2024-03/2024-03-17.md": "23f9672f5c93a6de52099b1b86834e8b", "set/directed-graph.md": "b4b8ad1be634a0a808af125fe8577a53", - "set/index.md": "b8165da42a81b5dc01b0a44ce365804e", + "set/index.md": "83f21533067c58ada1222a1c53ebbe8a", "set/graphs.md": "4bbcea8f5711b1ae26ed0026a4a69800", "_journal/2024-03-19.md": "a0807691819725bf44c0262405e97cbb", "_journal/2024-03/2024-03-18.md": "63c3c843fc6cfc2cd289ac8b7b108391", @@ -432,7 +432,7 @@ "_journal/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b", "_journal/2024-05/2024-05-12.md": "ca9f3996272152ef89924bb328efd365", "git/remotes.md": "2208e34b3195b6f1ec041024a66fb38b", - "programming/pred-trans.md": "fe30f0cab01fd31640f0778bf983747f", + "programming/pred-trans.md": "db73cc035e92cd019e7e6f79921e6c1e", "set/axioms.md": "063955bf19c703e9ad23be2aee4f1ab7", "_journal/2024-05-14.md": "f6ece1d6c178d57875786f87345343c5", "_journal/2024-05/2024-05-13.md": "71eb7924653eed5b6abd84d3a13b532b", @@ -446,15 +446,15 @@ "_journal/2024-05-17.md": "fb880d68077b655ede36d994554f3aba", "_journal/2024-05/2024-05-16.md": "9fdfadc3f9ea6a4418fd0e7066d6b10c", "_journal/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", - "hashing/direct-addressing.md": "17daf22ed3dfcc465924a175e8f11ce3", - "hashing/index.md": "340f8583eb51eaef011e3302bddb7ff8", - "set/classes.md": "3600fde1c4b30f600862d8d640962e4f", + "hashing/direct-addressing.md": "7ffaa27c01130d21aa32cf3b1c407785", + "hashing/index.md": "c870cf66e0224db58315ac0ba43b9cb1", + "set/classes.md": "bd3821b98c0893e1e13c06ff4def81ca", "_journal/2024-05-19.md": "fddd90fae08fab9bd83b0ef5d362c93a", "_journal/2024-05/2024-05-18.md": "c0b58b28f84b31cea91404f43b0ee40c", "_journal/2024-05/2024-05-17.md": "fb880d68077b655ede36d994554f3aba", "_journal/2024-05-20.md": "d58a4ecd3bf9621cbe688f043be61239", "_journal/2024-05/2024-05-19.md": "fc14fc23d4ddca3628df7eec71a07e27", - "_journal/2024-05-21.md": "4753ad41a519241d1ab7610bfe3c4038", + "_journal/2024-05-21.md": "7028d18a55f0a1f65dc0753af431ca42", "_journal/2024-05/2024-05-20.md": "d58a4ecd3bf9621cbe688f043be61239" }, "fields_dict": { diff --git a/notes/_journal/2024-05-21.md b/notes/_journal/2024-05-21.md index cacc73a..3994684 100644 --- a/notes/_journal/2024-05-21.md +++ b/notes/_journal/2024-05-21.md @@ -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 \ No newline at end of file +* 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. \ No newline at end of file diff --git a/notes/hashing/direct-addressing.md b/notes/hashing/direct-addressing.md index f415cac..af15f4b 100644 --- a/notes/hashing/direct-addressing.md +++ b/notes/hashing/direct-addressing.md @@ -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). - -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 ( 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). + +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 ( 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). + +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 ( 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). + +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). + +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). + +END%% + ## Bibliography * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). \ No newline at end of file diff --git a/notes/hashing/index.md b/notes/hashing/index.md index 0740643..b097cac 100644 --- a/notes/hashing/index.md +++ b/notes/hashing/index.md @@ -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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +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). + +END%% + +## Bibliography + +* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022). \ No newline at end of file diff --git a/notes/programming/pred-trans.md b/notes/programming/pred-trans.md index 26e2ee6..ca5f64f 100644 --- a/notes/programming/pred-trans.md +++ b/notes/programming/pred-trans.md @@ -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. 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. 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. 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. 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. 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. - -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 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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +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. + +END%% + ## Bibliography * Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. \ No newline at end of file diff --git a/notes/set/classes.md b/notes/set/classes.md index 3a59790..b445813 100644 --- a/notes/set/classes.md +++ b/notes/set/classes.md @@ -138,6 +138,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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). + +END%% + ## Russell's Paradox Let $R = \{x \mid x \not\in x\}$. Then $R \in R \Leftrightarrow R \not\in R$. diff --git a/notes/set/index.md b/notes/set/index.md index aa4e669..3589dbf 100644 --- a/notes/set/index.md +++ b/notes/set/index.md @@ -359,6 +359,82 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre 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)$$ + +%%ANKI +Basic +What does the union axiom (general form) state? +Back: For any set $A$, there exists a set $B$ whose elements are exactly the members of the members of $A$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the union axiom (general form) expressed using first-order logic? +Back: $$\forall A, \exists B, \forall x, x \in B \Leftrightarrow (\exists b \in B, x \in b)$$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What advantage does the general form of the union axiom have over its prelimiary form? +Back: The general form can handle infinite sets. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is the preliminary form of the union axiom proven using the general form? +Back: For any sets $a$ and $b$, $\bigcup \{a, b\} = a \cup b$. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the result of $\bigcup \{\{2, 4, 6\}, \{6, 16, 26\}, \{0\}\}$? +Back: $\{2, 4, 6, 16, 26, 0\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +What is the result of $\bigcup \varnothing$? +Back: $\varnothing$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +How is $\bigcup A$ represented in first-order logic? +Back: $\{x \mid \exists b \in A, x \in b\}$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +Roughly speaking, how does $\bigcup A$ adjust as $A$ gets larger? +Back: $\bigcup A$ gets larger. +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + +%%ANKI +Basic +If $A \subseteq B$, how do $\bigcup A$ and $\bigcup B$ relate? +Back: $\bigcup A \subseteq \bigcup B$ +Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977). + +END%% + ## Power Set Axiom For any set $a$, there is a set whose members are exactly the subsets of $a$: $$\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 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). + +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). + +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). + +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). + +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). + +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). + +END%% + +%%ANKI +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). + +END%% + +%%ANKI +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). + +END%% + ## Bibliography * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).