Left/right inverses. AoC.

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notes/_journal/2024-06-29.md

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 title: "2024-06-29"
 ---
 
-- [ ] Anki Flashcards
+- [x] Anki Flashcards
 - [x] KoL
 - [x] OGS
 - [ ] Sheet Music (10 min.)
 - [ ] Korean (Read 1 Story)
+
+* Left and right inverses. Touched on the Axiom of Choice.

notes/set/functions.md

@@ -9,7 +9,7 @@ tags:
 
 ## Overview
 
-A **function** $F$ is a single-valued [[relations|relation]]. We say $F$ **maps $A$ into $B$**, denoted $F \colon A \rightarrow B$, if and only if $F$ is a function, $\mathop{\text{dom}}A$, and $\mathop{\text{ran}}F \subseteq B$.
+A **function** $F$ is a single-valued [[relations|relation]]. We say $F$ **maps $A$ into $B$**, denoted $F \colon A \rightarrow B$, if and only if $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$.
 
 %%ANKI
 Basic
@@ -122,6 +122,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718464126883-->
 END%%
 
+%%ANKI
+Basic
+Is $\varnothing$ a function?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913529-->
+END%%
+
 ## Injections
 
 A function is **injective** or **one-to-one** if each element of the codomain is mapped to by at most one element of the domain.
@@ -258,6 +266,129 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718465870541-->
 END%%
 
+### Left Inverses
+
+Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a **left inverse**) such that $G \circ F = I_A$ if and only if $F$ is one-to-one.
+
+%%ANKI
+Basic
+What is the most specific mathematical object that describes a left inverse?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683931406-->
+END%%
+
+%%ANKI
+Basic
+How is a left inverse of $F \colon A \rightarrow B$ defined?
+Back: As a function $G \colon B \rightarrow A$ such that $G \circ F = I_A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719684322548-->
+END%%
+
+%%ANKI
+Basic
+How is a left inverse of set $A$ defined?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719684322553-->
+END%%
+
+%%ANKI
+Basic
+Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its domain?
+Back: $B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719680660507-->
+END%%
+
+%%ANKI
+Basic
+What does $I_A$ usually denote?
+Back: The identity function on set $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913532-->
+END%%
+
+%%ANKI
+Basic
+How is the identity function on set $B$ denoted?
+Back: $I_B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703723-->
+END%%
+
+%%ANKI
+Basic
+Consider $F \colon A \rightarrow B$. If $F$ has a left inverse, what is its codomain?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719680660511-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more simply write $G \circ F$?
+Back: $I_A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913534-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a left inverse of $F \colon A \rightarrow B$. How can we more simply write $F \circ G$?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913535-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ be a left inverse of function $G$. How do they interestingly compose?
+Back: As $F \circ G$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913538-->
+END%%
+
+%%ANKI
+Cloze
+Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:left} inverse iff $F$ is {3:one-to-one}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913540-->
+END%%
+
+%%ANKI
+Basic
+Does proving "left inverses iff injective" rely on AoC?
+Back: No.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913542-->
+END%%
+
+%%ANKI
+Basic
+What are the hypotheses of "left inverses iff injective"?
+Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703726-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. *Why* does "left inverses iff injective" require $A \neq \varnothing$?
+Back: Because a mapping from $B$ to $\varnothing$ cannot be a function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703729-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "left inverses iff injective" require AoC?
+Back: It doesn't.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703730-->
+END%%
+
 ## Surjections
 
 A function is **surjective** or **onto** if each element of the codomain is mapped to by at least one element of the domain. That is, **$F$ maps $A$ onto $B$** if and only if $F$ is a function, $\mathop{\text{dom}}A$, and $\mathop{\text{ran}}F = B$.
@@ -369,6 +500,120 @@ Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 202
 <!--ID: 1718465870573-->
 END%%
 
+### Right Inverses
+
+Assume that $F \colon A \rightarrow B$ is a function and $A \neq \varnothing$. Then there exists a function $G \colon B \rightarrow A$ (a right inverse) such that $F \circ G = I_B$ if and only if $F$ maps $A$ onto $B$.
+
+%%ANKI
+Basic
+What is the most specific mathematical object that describes a right inverse?
+Back: A function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683931410-->
+END%%
+
+%%ANKI
+Basic
+How is a right inverse of $F \colon A \rightarrow B$ defined?
+Back: As a function $G \colon B \rightarrow A$ such that $F \circ G = I_B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719684322557-->
+END%%
+
+%%ANKI
+Basic
+How is a right inverse of set $A$ defined?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719684322561-->
+END%%
+
+%%ANKI
+Cloze
+{1:Left} inverses are to {2:injections} whereas {2:right} inverses are to {1:surjections}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913533-->
+END%%
+
+%%ANKI
+Basic
+Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its domain?
+Back: $B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719680660514-->
+END%%
+
+%%ANKI
+Basic
+Consider $F \colon A \rightarrow B$. If $F$ has a right inverse, what is its codomain?
+Back: $A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719680660513-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more simply write $G \circ F$?
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913536-->
+END%%
+
+%%ANKI
+Basic
+Let $G$ be a right inverse of $F \colon A \rightarrow B$. How can we more simply write $F \circ G$?
+Back: The identity function on $B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913537-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ be a right inverse of function $G$. How do they interestingly compose?
+Back: As $G \circ F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913539-->
+END%%
+
+%%ANKI
+Cloze
+Suppose $F \colon A \rightarrow B$ and {1:$A \neq \varnothing$}. $F$ has a {2:right} inverse iff $F$ is {3:onto $B$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913541-->
+END%%
+
+%%ANKI
+Basic
+Does proving "right inverses iff surjective" rely on AoC?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913543-->
+END%%
+
+%%ANKI
+Basic
+What are the hypotheses of "right inverses iff surjective"?
+Back: Suppose $F \colon A \rightarrow B$ such that $A \neq \varnothing$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703732-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$. *Why* does "right inverses iff surjective" require $A \neq \varnothing$?
+Back: Because a mapping from $B$ to $\varnothing$ cannot be a function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703734-->
+END%%
+
+%%ANKI
+Basic
+Let $F \colon A \rightarrow B$ and $A \neq \varnothing$. *Why* does "right inverses iff surjective" require AoC?
+Back: There is no other mechanism for choosing an $x \in A$ for *each* $y \in B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683703736-->
+END%%
+
 ## Bijections
 
 A function is **bijective** or a **one-to-one correspondence** if each element of the codomain is mapped to by exactly one element of the domain.
@@ -441,6 +686,14 @@ END%%
 ## Inverses
 
 Let $F$ be an arbitrary set. The **inverse** of $F$ is the set $$F^{-1} = \{\langle u, v \rangle \mid vFu\}$$
+%%ANKI
+Basic
+What is the most specific mathematical object that describes an inverse?
+Back: A relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719683931414-->
+END%%
+
 %%ANKI
 Basic
 What kind of mathematical object does the inverse operation apply to?
@@ -656,6 +909,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1719398756558-->
 END%%
 
+%%ANKI
+Basic
+Consider function $F \colon \varnothing \rightarrow B$. What is $F^{-1}$?
+Back: $\varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913530-->
+END%%
+
 ## Compositions
 
 Let $F$ and $G$ be arbitrary sets. The **composition** of $F$ and $G$ is the set $$F \circ G = \{\langle u, v \rangle \mid \exists t, uGt \land tFv \}$$
@@ -997,14 +1258,6 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--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 $F[\![\varnothing]\!]$?

notes/set/index.md

@@ -850,6 +850,28 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717368558164-->
 END%%
 
+## Axiom of Choice
+
+### Relation Form
+
+For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$.
+
+%%ANKI
+Basic
+What is AoC an acronym for?
+Back: The **A**xiom **o**f **C**hoice.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913526-->
+END%%
+
+%%ANKI
+Basic
+What does the Axiom of Choice (relation form) state?
+Back: For any relation $R$ there exists a function $F \subseteq R$ with $\mathop{\text{dom}}F = \mathop{\text{dom}}R$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913527-->
+END%%
+
 ## Bibliography
 
 * Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).

notes/set/relations.md

@@ -95,6 +95,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717678753145-->
 END%%
 
+%%ANKI
+Basic
+Is $\varnothing$ a relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1719681913524-->
+END%%
+
 A **relation** $R$ is a set of ordered pairs. The **domain** of $R$ ($\mathop{\text{dom}}{R}$), the **range** of $R$ ($\mathop{\text{ran}}{R}$), and the **field** of $R$ ($\mathop{\text{fld}}{R}$) is defined as:
 
 * $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$