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notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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"Basic": [

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---
title: "2024-12-30"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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---
title: "2024-12-29"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Nore notes on cardinal number ordering.
* Preliminary notes on [[rdf#SPARQL|SPARQL]].

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@ -516,7 +516,7 @@ END%%
%%ANKI
Basic
What is the sign of the following bit-field?
What is the signedness of the following bit-field?
```c
struct foo { int bar : 1; };
```
@ -527,7 +527,7 @@ END%%
%%ANKI
Basic
What is the sign of the following bit-field?
What is the signedness of the following bit-field?
```c
struct foo { signed bar : 1; };
```
@ -538,7 +538,7 @@ END%%
%%ANKI
Basic
What is the sign of the following bit-field?
What is the signedness of the following bit-field?
```c
struct foo { unsigned bar : 1; };
```

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@ -229,7 +229,7 @@ END%%
%%ANKI
Basic
What is the linkage of a local variable without a storage specified?
What is the linkage of a local variable without a storage specifier?
Back: None.
Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf).
<!--ID: 1733010193297-->

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notes/c17/storage.md
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@ -298,7 +298,7 @@ END%%
%%ANKI
Basic
Which storage class specifier(s) can be used to declare an object with automatic storage duration?
Back: `auto` and {`register`}.
Back: `auto` and `register`.
Reference: “ISO: Programming Languages - C17,” April 2017, [https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf](https://www.open-std.org/jtc1/sc22/wg14/www/abq/c17_updated_proposed_fdis.pdf).
<!--ID: 1733064332819-->
END%%

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notes/calculus/integrals.md
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@ -159,7 +159,7 @@ END%%
%%ANKI
Basic
Let $s$ be a step function over $[a, b]$. What does $\int_a^b s(x) \,dx$ evaluate to after swapping limits of integration?
Back: As $-\int_b^a s(x) \,dx$.
Back: $-\int_b^a s(x) \,dx$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734816555515-->
END%%
@ -202,7 +202,7 @@ Let $s$ and $t$ be step functions defined on $[a, b]$. Then $$\int_a^b s(x) + t(
%%ANKI
Basic
Let $s$ and $t$ be step functions over $[a, b]$. What does the additive property state?
Let $s$ and $t$ be step functions over $[a, b]$. What does the additive property of integrals state?
Back: $\int_a^b s(x) + t(x) \,dx = \int_a^b s(x) \,dx + \int_a^b t(x) \,dx$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734814463659-->
@ -231,7 +231,7 @@ Let $s$ be a step function defined on $[a, b]$. Let $c \in \mathbb{R}$. Then $$\
%%ANKI
Basic
Let $s$ be a step function over $[a, b]$. What does the homogeneous property state?
Let $s$ be a step function over $[a, b]$. What does the homogeneous property of integrals state?
Back: For all $c \in \mathbb{R}$, $\int_a^b c \cdot s(x) \,dx = c \int_a^b s(x) \,dx$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734814463679-->
@ -252,7 +252,7 @@ Let $s$ and $t$ be step functions defined on $[a, b]$. Let $c_1, c_2 \in \mathbb
%%ANKI
Basic
Let $s$ and $t$ be step functions over $[a, b]$ and $c_1, c_2 \in \mathbb{R}$. What does the linearity property state?
Let $s$ and $t$ be step functions over $[a, b]$ and $c_1, c_2 \in \mathbb{R}$. What does the linearity property of integrals state?
Back: $\int_a^b [c_1 s(x) + c_2 t(x)] \,dx = c_1 \int_a^b s(x) \,dx + c_2 \int_a^b t(x) \,dx$
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734814463699-->
@ -276,7 +276,7 @@ END%%
%%ANKI
Cloze
The {linearity} property is a combination of the {additive} and {homogenous} properties.
The {linearity} property of integrals is a combination of the {additive} and {homogenous} properties.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734814463693-->
END%%
@ -287,8 +287,8 @@ Let $s$ and $t$ be step functions defined on $[a, b]$. Suppose $s(x) < t(x)$ for
%%ANKI
Basic
Let $s$ and $t$ be step functions over $[a, b]$. What does the comparison theorem state?
Back: If $s(x) < t(x)$ for all $x \in [a, b]$, $\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$.
Let $s$ and $t$ be step functions over $[a, b]$. What does the comparison theorem of integrals state?
Back: If $s(x) < t(x)$ for all $x \in [a, b]$, then $\int_a^b s(x) \,dx < \int_a^b t(x) \,dx$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1734815755275-->
END%%

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@ -183,14 +183,6 @@ Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third ed
<!--ID: 1734903366666-->
END%%
%%ANKI
Basic
How is the concatenation of strings $x$ and $y$ over some alphabet denoted?
Back: $xy$
Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
<!--ID: 1734903366671-->
END%%
%%ANKI
Basic
Let $x$ be a string over some alphabet. What does $xx$ denote?
@ -201,7 +193,7 @@ END%%
%%ANKI
Cloze
Let $x$ be a string over some alphabet. Then {$xx$} $=$ {$x^2$}.
Let $x$ be a string over some alphabet. Then {$xx$} is equivalently denoted as {$x^2$}.
Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
<!--ID: 1734903366679-->
END%%
@ -249,7 +241,7 @@ END%%
%%ANKI
Basic
What is a language?
Back: A nonempty set of strings.
Back: A set of strings.
Reference: Michael Sipser, _Introduction to the Theory of Computation_, Third edition, international edition (Australia Brazil Japan Korea Mexiko Singapore Spain United Kingdom United States: Cengage Learning, 2013).
<!--ID: 1734903366705-->
END%%

232
notes/data-models/rdf.md
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@ -126,7 +126,7 @@ END%%
%%ANKI
Basic
RDF identifiers use what encoding?
Back: URIs.
Back: IRIs.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1722191359882-->
END%%
@ -142,7 +142,7 @@ END%%
%%ANKI
Basic
How does RDF define a named graph?
Back: A graph with a name.
Back: As a graph (a set of triples) with a name.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1734804443462-->
END%%
@ -170,21 +170,33 @@ Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web
<!--ID: 1734804443468-->
END%%
The RDF data model was designed from the beginning with [[federation|data federation]] in mind. Because every RDF store represents RDF data in the same way (i.e. as a set of triples), federation is as simple as combining the commonly encoded data into a single store before querying.
%%ANKI
Cloze
RDF is a {data model} whereas XML is a {serialization}.
Basic
What mechanism makes RDF data federation easy?
Back: Every source communicates data in the same way (i.e. as a set of triples).
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: serialization::xml
<!--ID: 1734805690321-->
Tags: data_model::federation
<!--ID: 1735176997772-->
END%%
%%ANKI
Basic
How might RDF and XML relate to one another?
Back: XML can be used to serialize RDF.
In the context of RDF, what does federation refer to?
Back: The combining of multiple RDF stores into a single (possibly virtual) view.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: serialization::xml
<!--ID: 1734805690328-->
Tags: data_model::federation
<!--ID: 1735176997776-->
END%%
%%ANKI
Basic
What does the federated graph refer to?
Back: The graph produced by combining various federated data sources.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: data_model::federation
<!--ID: 1735176997779-->
END%%
## Blank Nodes
@ -239,37 +251,203 @@ Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web
<!--ID: 1735162429102-->
END%%
## Federation
## Serializations
The RDF data model was designed from the beginning with [[federation|data federation]] in mind. Because every RDF store represents RDF data in the same way (i.e. as a set of triples), federation is as simple as combining the commonly encoded data into a single store before querying.
The original syntax and standard for writing RDF was RDF/XML. Turtle is an alternative, more human-readable, syntax. Besides these two exist many others.
%%ANKI
Basic
What mechanism makes RDF data federation easy?
Back: Every source communicates data in the same way (i.e. as a set of triples).
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: data_model::federation
<!--ID: 1735176997772-->
Which RDF serialization is most widely used?
Back: Turtle.
Reference: “Resource Description Framework,” in _Wikipedia_, November 25, 2024, [https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework](https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework&oldid=1259544587).
<!--ID: 1735507470222-->
END%%
%%ANKI
Basic
In the context of RDF, what does federation refer to?
Back: The combining of multiple RDF stores into a single (possibly virtual) view.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: data_model::federation
<!--ID: 1735176997776-->
Why is Turtle named the way it is?
Back: It is an "acronym" for **Te**rse **R**DF **T**riple **L**anguag**e**.
Reference: “Resource Description Framework,” in _Wikipedia_, November 25, 2024, [https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework](https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework&oldid=1259544587).
<!--ID: 1735507470227-->
END%%
%%ANKI
Basic
What does the federated graph refer to?
Back: The graph produced by combining various federated data sources.
What is RDF/XML a container for?
Back: RDF data.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: data_model::federation
<!--ID: 1735176997779-->
<!--ID: 1735507470230-->
END%%
%%ANKI
Basic
What is Turtle a container for?
Back: RDF data.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470233-->
END%%
%%ANKI
Cloze
RDF is a {data model} whereas XML is a {serialization format}.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: serialization::xml
<!--ID: 1734805690321-->
END%%
%%ANKI
Cloze
RDF is a {data model} whereas Turtle is a {serialization format}.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470236-->
END%%
%%ANKI
Basic
How might RDF and XML relate to one another?
Back: XML can be used to serialize RDF.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
Tags: serialization::xml
<!--ID: 1734805690328-->
END%%
## SPARQL
SPARQL (SPARQL Protocol and RDF Query Language) is the standard way to access RDF data using a query language. Query patterns are represented in a variant of Turtle, the most ubiquitous language used to express RDF itself.
A server for the SPARQL protocol is called a **SPARQL endpoint**. It accepts SPARQL queries and returns results, according to the details of the protocol.
%%ANKI
Basic
SPARQL syntax most closely resembles which RDF serialization format?
Back: Turtle.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470239-->
END%%
%%ANKI
Basic
What is the standard means of querying an RDF store?
Back: SPARQL.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488320-->
END%%
%%ANKI
Basic
What is SPARQL an acronym for?
Back: **S**PARQL **P**rotocol **a**nd **R**DF **Q**uery **L**anguage.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488325-->
END%%
%%ANKI
Cloze
{1:SPARQL} is to {2:RDF} as {2:SQL} is to {1:RDBMS}.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488332-->
END%%
%%ANKI
Basic
The following snippet is an example of what language?
```sparql
SELECT ?movie WHERE {:JamesDean :playedIn ?movie}
```
Back: SPARQL
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488335-->
END%%
%%ANKI
Basic
What do the `:`s indicate in the following query?
```sparql
SELECT ?movie WHERE {:JamesDean :playedIn ?movie}
```
Back: The default namespace.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488338-->
END%%
%%ANKI
Basic
How are namespaces introduced in a SPARQL query?
Back: With the `:` character.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470242-->
END%%
%%ANKI
Basic
What do the `?`s indicate in the following query?
```sparql
SELECT ?movie WHERE {:JamesDean :playedIn ?movie}
```
Back: Variables to be matched against.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488341-->
END%%
%%ANKI
Basic
How are variables introduced in a SPARQL query?
Back: With the `?` character.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470245-->
END%%
%%ANKI
Basic
What do the curly braces (`{}`) indicate in the following query?
```sparql
SELECT ?movie WHERE {:JamesDean :playedIn ?movie}
```
Back: A graph pattern.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735506488344-->
END%%
%%ANKI
Basic
How are graph patterns introduced in a SPARQL query?
Back: With the `{` and `}` characters.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470248-->
END%%
%%ANKI
Basic
In SPARQL, what is a graph pattern?
Back: A graph with wildcards, used to match against a data graph.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735521898594-->
END%%
%%ANKI
Cloze
A {server for the SPARQL protocol} is called a {SPARQL endpoint}.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470252-->
END%%
%%ANKI
Basic
What is a SPARQL endpoint?
Back: A server that can respond to SPARQL queries.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470256-->
END%%
%%ANKI
Basic
What does Allemang et al. describe as the most web-friendly way to provide access to RDF data?
Back: Exposing a SPARQL endpoint to the data.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1735507470259-->
END%%
## Bibliography
* Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
* Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
* “Resource Description Framework,” in _Wikipedia_, November 25, 2024, [https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework](https://en.wikipedia.org/w/index.php?title=Resource_Description_Framework&oldid=1259544587).

17
notes/linkers/elf.md
View File

@ -198,7 +198,7 @@ Thread-local global and static C variables initialized to a non-zero value.
%%ANKI
Basic
What kind of global/static C variables does the `.tdata` section contain?
What kind of global and static C variables does the `.tdata` section contain?
Back: Thread-local variables initialized to a non-zero value.
Reference: Ulrich Drepper, “ELF Handling For Thread-Local Storage,” n.d.
Tags: c17
@ -395,7 +395,7 @@ Uninitialized global and static thread-local C variables, along with any global
%%ANKI
Basic
What kind of global/static C variables does the `.tbss` section contain?
What kind of global and static C variables does the `.tbss` section contain?
Back: Unitialized thread-local variables or those initialized to zero.
Reference: Ulrich Drepper, “ELF Handling For Thread-Local Storage,” n.d.
Tags: c17
@ -422,8 +422,8 @@ END%%
%%ANKI
Basic
Which two ELF sections contain global/static thread-local C variables?
Back: `.tdata` and `.bss`
Which two ELF sections contain global and static thread-local C variables?
Back: `.tdata` and `.tbss`
Reference: Ulrich Drepper, “ELF Handling For Thread-Local Storage,” n.d.
Tags: c17
<!--ID: 1734745402975-->
@ -471,15 +471,6 @@ Tags: c17
<!--ID: 1734745402993-->
END%%
%%ANKI
Basic
How does the size of the `.tbss` section compare to that of the `.tdata` section?
Back: Less than or equal since the `.tbss` section is always empty.
Reference: Ulrich Drepper, “ELF Handling For Thread-Local Storage,” n.d.
Tags: c17
<!--ID: 1734745402997-->
END%%
%%ANKI
Basic
Consider the following translation unit. Which ELF section will `bar` end up in?

172
notes/set/cardinality.md
View File

@ -1238,7 +1238,9 @@ END%%
### Ordering
A set $A$ is **dominated** by a set $B$, written $A \preceq B$, if and only if there is a one-to-one function from $A$ into $B$. In other words, $A \preceq B$ if and only if $A$ is equinumerous to some subset of $B$.
A set $A$ is **dominated** by a set $B$, written $A \preceq B$, if and only if there is a one-to-one function from $A$ into $B$. In other words, $A \preceq B$ if and only if $A$ is equinumerous to some subset of $B$. Then $$\mathop{\text{card}}A \leq \mathop{\text{card}}B \text{ if and only if } A \preceq B.$$
Furthermore, $$\mathop{\text{card}}A < \mathop{\text{card}}B \text{ if and only if } A \preceq B \text{ and } A \not\approx B.$$
%%ANKI
Basic
@ -1248,6 +1250,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1735353438914-->
END%%
%%ANKI
Basic
How do we denote that set $A$ is strictly dominated by set $B$?
Back: $A \prec B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305902-->
END%%
%%ANKI
Basic
Suppose $A \preceq B$. Then what must exist by definition?
@ -1256,14 +1266,46 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1735353438921-->
END%%
%%ANKI
Basic
Suppose $A \prec B$. Then what must exist by definition?
Back: A one-to-one function from $A$ into $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305907-->
END%%
%%ANKI
Basic
Suppose $A \preceq B$. Then what must $A$ be equinumerous to?
Back: Some subset of $B$.
Back: A subset of $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735353438924-->
END%%
%%ANKI
Basic
Suppose $A \preceq B$. Then what must $A$ *not* be equinumerous to?
Back: N/A. There is no restriction here.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305910-->
END%%
%%ANKI
Basic
Suppose $A \prec B$. Then what must $A$ be equinumerous to?
Back: A subset of $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305914-->
END%%
%%ANKI
Basic
Suppose $A \prec B$. Then what must $A$ *not* be equinumerous to?
Back: $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305917-->
END%%
%%ANKI
Basic
What does $A \preceq B$ denote?
@ -1272,6 +1314,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1735353438928-->
END%%
%%ANKI
Basic
What does $A \prec B$ denote?
Back: That $A$ is strictly dominated by $B$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305921-->
END%%
%%ANKI
Basic
How do we expand expression $A \preceq B$ using FOL?
@ -1280,6 +1330,124 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1735353438932-->
END%%
%%ANKI
Basic
How do we expand expression $A \prec B$ using FOL?
Back: $A \not\approx B \land \exists C, C \subseteq B \land A \approx C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305924-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. How is $\kappa \leq \lambda$ defined?
Back: As $K \preceq L$ for sets satisfying $\mathop{\text{card}}K = \kappa$ and $\mathop{\text{card}} L = \lambda$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305927-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. How is $\kappa < \lambda$ defined?
Back: As $K \preceq L$ and $K \not\approx L$ for sets satisfying $\mathop{\text{card}}K = \kappa$ and $\mathop{\text{card}} L = \lambda$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305931-->
END%%
%%ANKI
Cloze
{$\leq$} on cardinal numbers corresponds to {$\preceq$} on sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305934-->
END%%
%%ANKI
Cloze
{$<$} on cardinal numbers corresponds to {$\prec$} on sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305942-->
END%%
%%ANKI
Basic
How is $\mathop{\text{card} }K \leq \mathop{\text{card} }L$ defined in terms of equinumerosity?
Back: $\mathop{\text{card} }K \leq \mathop{\text{card} }L$ iff $K$ is equinumerous to a subset of $L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305947-->
END%%
%%ANKI
Basic
How is $\mathop{\text{card} }K < \mathop{\text{card} }L$ defined in terms of equinumerosity?
Back: $\mathop{\text{card} }K < \mathop{\text{card} }L$ iff $K$ is equinumerous to a subset of $L$ and $K \not\approx L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305952-->
END%%
%%ANKI
Basic
Let $\kappa$ and $\lambda$ be cardinal numbers. Restate the following in terms of sets: $$\kappa < \lambda \text{ iff } \kappa \leq \lambda \text{ and } \kappa \neq \lambda$$
Back: Given $\mathop{\text{card}}K = \kappa$ and $\mathop{\text{card}}L = \lambda$, $\mathop{\text{card}}K < \mathop{\text{card}}L$ iff $K \preceq L$ and $K \not\approx L$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305957-->
END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. Restate the following in terms of cardinal numbers: $$\mathop{\text{card}}K < \mathop{\text{card}}L \text{ iff } K \preceq L \text{ and } K \not\approx L.$$
Back: Given $\mathop{\text{card}}K = \kappa$ and $\mathop{\text{card}}L = \lambda$, $\kappa < \lambda$ iff $\kappa \leq \lambda$ and $\kappa \neq \lambda$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493305962-->
END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. *Why* can't we use the following definition? $$\mathop{\text{card}} K \leq \mathop{\text{card}} L \text{ iff } \exists A \subseteq L, K \approx A$$
Back: N/A. This is a suitable definition.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493767848-->
END%%
%%ANKI
Basic
Let $K$ and $L$ be sets. *Why* can't we use the following definition? $$\mathop{\text{card}} K < \mathop{\text{card}} L \text{ iff } \exists A \subset L, K \approx A$$
Back: Infinite sets may be equinumerous to a proper subset of themselves.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735493767854-->
END%%
%%ANKI
Basic
For any $n \in \omega$, *why* is $n < \aleph_0$?
Back: $n \not\approx \omega$ and there exists an injective function $f \colon n \rightarrow \omega$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735522988804-->
END%%
%%ANKI
Basic
For any cardinal number $\kappa$, *why* is $\kappa < 2^\kappa$?
Back: Assuming $\mathop{\text{card}}K = \kappa$, $K \not\approx \mathscr{P}(K)$ and there exists an injective function $f \colon K \rightarrow \mathscr{P}(K)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735522988810-->
END%%
%%ANKI
Basic
What is the smallest cardinal number?
Back: $0$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735522988813-->
END%%
%%ANKI
Basic
What is the largest cardinal number?
Back: N/A. There is no largest cardinal number.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1735522988817-->
END%%
## Bibliography
* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).