From 6c9e34e19209e0bc3bb850367afa46a8f31dfcc6 Mon Sep 17 00:00:00 2001
From: Joshua Potter <jrpotter2112@gmail.com>
Date: Sat, 15 Jun 2024 09:38:37 -0600
Subject: [PATCH] Expand notes on function categories.

---
 .../plugins/obsidian-to-anki-plugin/data.json |  21 +-
 notes/_journal/2024-06-15.md                  |  11 +
 notes/_journal/{ => 2024-06}/2024-06-13.md    |   0
 notes/_journal/2024-06/2024-06-14.md          |  12 +
 notes/algebra/sequences/delta-constant.md     |   2 +-
 notes/lambda-calculus/alpha-conversion.md     |  85 +++-
 notes/logic/equiv-trans.md                    |   2 +-
 notes/set/functions.md                        | 444 ++++++++++++++++++
 notes/set/images/function-bijective.png       | Bin 0 -> 3379 bytes
 notes/set/images/function-general.png         | Bin 0 -> 4020 bytes
 notes/set/images/function-injective.png       | Bin 0 -> 3276 bytes
 notes/set/images/function-surjective.png      | Bin 0 -> 3332 bytes
 notes/set/relations.md                        |  26 +
 13 files changed, 588 insertions(+), 15 deletions(-)
 create mode 100644 notes/_journal/2024-06-15.md
 rename notes/_journal/{ => 2024-06}/2024-06-13.md (100%)
 create mode 100644 notes/_journal/2024-06/2024-06-14.md
 create mode 100644 notes/set/functions.md
 create mode 100644 notes/set/images/function-bijective.png
 create mode 100644 notes/set/images/function-general.png
 create mode 100644 notes/set/images/function-injective.png
 create mode 100644 notes/set/images/function-surjective.png

diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
index d832889..04a38d1 100644
--- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
+++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
@@ -137,7 +137,11 @@
     "venn-diagram-symm-diff.png",
     "relation-ordering-example.png",
     "open-addressing.png",
-    "closed-addressing.png"
+    "closed-addressing.png",
+    "function-bijective.png",
+    "function-injective.png",
+    "function-surjective.png",
+    "function-general.png"
   ],
   "File Hashes": {
     "algorithms/index.md": "3ac071354e55242919cc574eb43de6f8",
@@ -176,7 +180,7 @@
     "_journal/2024-02-02.md": "a3b222daee8a50bce4cbac699efc7180",
     "_journal/2024-02-01.md": "3aa232387d2dc662384976fd116888eb",
     "_journal/2024-01-31.md": "7c7fbfccabc316f9e676826bf8dfe970",
-    "logic/equiv-trans.md": "fb7f2027b2b323374580fde8a1de579e",
+    "logic/equiv-trans.md": "c52d0907d35d7a3c2e4576f2bd411257",
     "_journal/2024-02-07.md": "8d81cd56a3b33883a7706d32e77b5889",
     "algorithms/loop-invariants.md": "cbefc346842c21a6cce5c5edce451eb2",
     "algorithms/loop-invariant.md": "3b390e720f3b2a98e611b49a0bb1f5a9",
@@ -377,7 +381,7 @@
     "_journal/2024-04-16.md": "0bf6e2f2a3afab73d528cee88c4c1a92",
     "_journal/2024-04/2024-04-15.md": "256253b0633d878ca58060162beb7587",
     "algebra/polynomials.md": "da56d2d6934acfa2c6b7b2c73c87b2c7",
-    "algebra/sequences/delta-constant.md": "70f45d7b8d5c3a147fabc279105c4983",
+    "algebra/sequences/delta-constant.md": "d9af958375cdf993e4ac3c68c1324ba7",
     "_journal/2024-04-19.md": "a293087860a7f378507a96df0b09dd2b",
     "_journal/2024-04/2024-04-18.md": "f6e5bee68dbef90a21ca92a846930a88",
     "_journal/2024-04/2024-04-17.md": "331423470ea83fc990c1ee1d5bd3b3f1",
@@ -503,12 +507,12 @@
     "_journal/2024-06/2024-06-04.md": "52b28035b9c91c9b14cef1154c1a0fa1",
     "_journal/2024-06-06.md": "3f9109925dea304e7172df39922cc95a",
     "_journal/2024-06/2024-06-05.md": "b06a0fa567bd81e3b593f7e1838f9de1",
-    "set/relations.md": "303f83287d33a300cf8c7fafe2834235",
+    "set/relations.md": "d486836acec494ea3b185ec9746df7c9",
     "_journal/2024-06-07.md": "795be41cc3c9c0f27361696d237604a2",
     "_journal/2024-06/2024-06-06.md": "db3407dcc86fa759b061246ec9fbd381",
     "_journal/2024-06-08.md": "b20d39dab30b4e12559a831ab8d2f9b8",
     "_journal/2024-06/2024-06-07.md": "c6bfc4c1e5913d23ea7828a23340e7d3",
-    "lambda-calculus/alpha-conversion.md": "c0d40271a14b1f44b937de7791ca089b",
+    "lambda-calculus/alpha-conversion.md": "9965a24624a745af16f10d9ffd78cc0c",
     "lambda-calculus/index.md": "756c93b8717fd00b04f8a99509066486",
     "x86-64/instructions/condition-codes.md": "56ad6eb395153609a1ec51835925e8c9",
     "x86-64/instructions/logical.md": "818428b9ef84753920dc61e5c2de9199",
@@ -526,7 +530,12 @@
     "hashing/open-addressing.md": "c27e92f2865bbb426fdd1e30fc52f1ed",
     "hashing/closed-addressing.md": "962a48517969bf5e410cf78fc584051f",
     "_journal/2024-06-13.md": "dec86b3a3e43eca306c3cf9a46b260ed",
-    "_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e"
+    "_journal/2024-06/2024-06-12.md": "f82dfa74d0def8c3179d3d076f94558e",
+    "_journal/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307",
+    "_journal/2024-06/2024-06-13.md": "e2722a00585d94794a089e8035e05728",
+    "set/functions.md": "34bf35a8ae16a0d735ce7e3e1b5bfa05",
+    "_journal/2024-06-15.md": "92cb8dc5c98e10832fb70c0e3ab3cec4",
+    "_journal/2024-06/2024-06-14.md": "5d12bc272238ac985a1d35d3d63ea307"
   },
   "fields_dict": {
     "Basic": [
diff --git a/notes/_journal/2024-06-15.md b/notes/_journal/2024-06-15.md
new file mode 100644
index 0000000..b9970e9
--- /dev/null
+++ b/notes/_journal/2024-06-15.md
@@ -0,0 +1,11 @@
+---
+title: "2024-06-15"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* [[functions|Notes]] on injections, surjections, and bijections.
\ No newline at end of file
diff --git a/notes/_journal/2024-06-13.md b/notes/_journal/2024-06/2024-06-13.md
similarity index 100%
rename from notes/_journal/2024-06-13.md
rename to notes/_journal/2024-06/2024-06-13.md
diff --git a/notes/_journal/2024-06/2024-06-14.md b/notes/_journal/2024-06/2024-06-14.md
new file mode 100644
index 0000000..a5ff0ea
--- /dev/null
+++ b/notes/_journal/2024-06/2024-06-14.md
@@ -0,0 +1,12 @@
+---
+title: "2024-06-14"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Finished most notes on [[alpha-conversion|α-conversion]].
+* Starting notes on [[functions]].
\ No newline at end of file
diff --git a/notes/algebra/sequences/delta-constant.md b/notes/algebra/sequences/delta-constant.md
index bd163f4..2a7d906 100644
--- a/notes/algebra/sequences/delta-constant.md
+++ b/notes/algebra/sequences/delta-constant.md
@@ -119,7 +119,7 @@ END%%
 
 %%ANKI
 Cloze
-The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences {is constant}.
+The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences are {constant}.
 Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 <!--ID: 1713580109237-->
 END%%
diff --git a/notes/lambda-calculus/alpha-conversion.md b/notes/lambda-calculus/alpha-conversion.md
index b20852b..03f7b5c 100644
--- a/notes/lambda-calculus/alpha-conversion.md
+++ b/notes/lambda-calculus/alpha-conversion.md
@@ -152,6 +152,8 @@ Let $x$, $y$, and $v$ be distinct variables. Then
 * $v \not\in FV(M) \Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$
 * $v \not\in FV(M) \Rightarrow [x/v][v/x]M \equiv_\alpha M$
 * $y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
+* $x \not\in FV(Q) \land y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$
+* $[P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
 
 %%ANKI
 Basic
@@ -186,9 +188,8 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 END%%
 
 %%ANKI
-Basic
-$[P/v][v/x]M \equiv_\alpha [P/x]M$ is necessary for what condition?
-Back: $v \not\in FV(M)$
+Cloze
+{$v \not\in FV(M)$} $\Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$
 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: 1717855810777-->
 END%%
@@ -266,16 +267,53 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 END%%
 
 %%ANKI
-Basic
-$[P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$ is necessary for what condition?
-Back: $y \not\in FV(P)$
+Cloze
+{$y \not\in FV(P)$} $\Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
 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: 1717855810784-->
 END%%
 
+%%ANKI
+Cloze
+{$x \not\in FV(Q) \land y \not\in FV(P)$} $\Rightarrow [P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$
+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: 1718422235903-->
+END%%
+
 %%ANKI
 Basic
-What happens if the antecedent is false in $y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$?
+$[P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$ is a specialization of what more general congruence?
+Back: $[P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
+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: 1718422235909-->
+END%%
+
+%%ANKI
+Cloze
+{$F$} $\Rightarrow [P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
+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: 1718422235912-->
+END%%
+
+%%ANKI
+Basic
+What expression containing nested substitutions is congruent to $[P/x][Q/x]M$?
+Back: $[([P/x]Q)/x]M$
+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: 1718422235916-->
+END%%
+
+%%ANKI
+Basic
+What expression containing adjacent substitutions is congruent to $[([P/x]Q)/x]M$?
+Back: $[P/x][Q/x]M$
+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: 1718422279995-->
+END%%
+
+%%ANKI
+Basic
+What happens if the antecedent of the following lemma is false? $$y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$$
 Back: $y$ is subbed in $M$ on the LHS but subbed in both $P$ and $M$ on the RHS.
 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: 1717855810787-->
@@ -313,6 +351,39 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
 <!--ID: 1717855810802-->
 END%%
 
+For $\lambda$-terms $M$, $M'$, $N$, and $N'$, and variable $x$, $$M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$$
+
+%%ANKI
+Basic
+The proof of which implication shows "substitution is well-behaved w.r.t. $\alpha$-conversion"?
+Back: $M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$
+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: 1718422973129-->
+END%%
+
+%%ANKI
+Basic
+What does Hindley et al. mean by "substitution is well-behaved w.r.t. $\alpha$-conversion"?
+Back: $\alpha$-converting substitution inputs yields congruent outputs.
+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: 1718422973135-->
+END%%
+
+%%ANKI
+Cloze
+{$M \equiv_\alpha M' \land N \equiv_\alpha N'$} $\Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$
+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: 1718422973141-->
+END%%
+
+%%ANKI
+Basic
+What does Hindley et al. say the following implication says about substitution? $$M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$$
+Back: It is well-defined with respect to $\alpha$-conversion.
+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: 1718422981125-->
+END%%
+
 ## Bibliography
 
 * 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).
\ No newline at end of file
diff --git a/notes/logic/equiv-trans.md b/notes/logic/equiv-trans.md
index 3b5fed8..9ce5ed1 100644
--- a/notes/logic/equiv-trans.md
+++ b/notes/logic/equiv-trans.md
@@ -506,7 +506,7 @@ END%%
 %%ANKI
 Basic
 If $x \neq e$, why might $E_e^x = E$ be an equivalence despite $x$ existing in $E$?
-Back: If the only occurrences of $x$ in $E$ are bound.
+Back: The only occurrences of $x$ in $E$ may be bound.
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1707762304135-->
 END%%
diff --git a/notes/set/functions.md b/notes/set/functions.md
new file mode 100644
index 0000000..6ff9d2d
--- /dev/null
+++ b/notes/set/functions.md
@@ -0,0 +1,444 @@
+---
+title: Functions
+TARGET DECK: Obsidian::STEM
+FILE TAGS: set::function
+tags:
+  - function
+  - set
+---
+
+## 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 $\mathop{\text{ran}}F \subseteq B$.
+
+%%ANKI
+Basic
+Which of relations or functions is the more general concept?
+Back: Relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443345-->
+END%%
+
+%%ANKI
+Basic
+What *is* a function?
+Back: A relation $F$ such that for each $x \in \mathop{\text{dom}}F$, there exists a unique $y$ such that $xFy$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443366-->
+END%%
+
+%%ANKI
+Basic
+For function $F$ and $x \in \mathop{\text{dom}}F$, what name is given to $F(x)$?
+Back: The value of $F$ at $x$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443370-->
+END%%
+
+%%ANKI
+Basic
+Who introduced the function notation $F(x)$?
+Back: Leonhard Euler.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443374-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $y$.
+Back: $\langle x, F(x) \rangle \in F$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443379-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ be a function and $\langle x, y \rangle \in F$. Rewrite the membership as an expression excluding $x$.
+Back: N/A.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443384-->
+END%%
+
+%%ANKI
+Basic
+Consider notation $F(x)$. What assumption is $F$ assumed to satisfy?
+Back: It is assumed to be a function.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443388-->
+END%%
+
+%%ANKI
+Basic
+Consider notation $F(x)$. What assumption is $x$ assumed to satisfy?
+Back: It is assumed to be in the domain of $F$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443393-->
+END%%
+
+%%ANKI
+Cloze
+A function is a {single-valued} relation.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443397-->
+END%%
+
+%%ANKI
+Basic
+How is $F \colon A \rightarrow B$ pronounced?
+Back: $F$ maps $A$ into $B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443401-->
+END%%
+
+%%ANKI
+Basic
+What three conditions hold iff $F$ maps $A$ into $B$?
+Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F \subseteq B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443404-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. What term is used to refer to $A$?
+Back: The domain.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718464126872-->
+END%%
+
+%%ANKI
+Basic
+Consider function $F \colon A \rightarrow B$. What term is used to refer to $B$?
+Back: The codomain.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718464126879-->
+END%%
+
+%%ANKI
+Basic
+How does the range of a function compare to its codomain?
+Back: The range is a subset of the codomain.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718464126883-->
+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.
+
+%%ANKI
+Basic
+What does it mean for a function to be injective?
+Back: Each element of the codomain is mapped to by at most one element of the domain.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464126887-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a function to be one-to-one?
+Back: Each element of the codomain is mapped to by at most one element of the domain.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870487-->
+END%%
+
+%%ANKI
+Basic
+Each element of an injection's codomain is mapped to by how many elements of the domain?
+Back: At most one.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464498595-->
+END%%
+
+%%ANKI
+Basic
+Suppose `Function.Injective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$?
+Back: $\forall a_1, a_2 \in A, (f(a_1) = f(a_2) \Rightarrow a_1 = a_2$)
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+Tags: lean logic::predicate
+<!--ID: 1718464498603-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict an injection?
+![[function-bijective.png]]
+Back: Yes.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870490-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict a one-to-one function?
+![[function-injective.png]]
+Back: Yes.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870493-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict a one-to-one function?
+![[function-surjective.png]]
+Back: No.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870497-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't the following an injection?
+![[function-general.png]]
+Back: Both $1 \mapsto d$ and $2 \mapsto d$.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870505-->
+END%%
+
+%%ANKI
+Basic
+Is a single-valued set a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443358-->
+END%%
+
+%%ANKI
+Basic
+Is a single-valued relation a function?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443362-->
+END%%
+
+%%ANKI
+Basic
+Is a single-rooted set a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870509-->
+END%%
+
+%%ANKI
+Basic
+Is a single-rooted relation a function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870519-->
+END%%
+
+%%ANKI
+Cloze
+{One-to-one} is to functions whereas {single-rooted} is to relations.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870525-->
+END%%
+
+%%ANKI
+Basic
+Is a one-to-one function a single-rooted relation?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870531-->
+END%%
+
+%%ANKI
+Basic
+Is a single-rooted relation a one-to-one function?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870536-->
+END%%
+
+%%ANKI
+Basic
+Is a single-rooted function a one-to-one function?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870541-->
+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 $\mathop{\text{ran}}F = B$.
+
+%%ANKI
+Basic
+What does it mean for function to be surjective?
+Back: Each element of the codomain is mapped to by at least one element of the domain.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464126891-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a function to be onto?
+Back: Each element of the codomain is mapped to by at least one element of the domain.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870546-->
+END%%
+
+%%ANKI
+Basic
+Each element of a surjection's codomain is mapped to by how many elements of the domain?
+Back: At least one.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464498606-->
+END%%
+
+%%ANKI
+Basic
+Suppose `Function.Surjective f` for $f \colon A \rightarrow B$. What predicate logical formula describes $f$?
+Back: $\forall b \in B, \exists a \in A, f(a) = b$
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+Tags: lean logic::predicate
+<!--ID: 1718464498615-->
+END%%
+
+%%ANKI
+Cloze
+{1:Injective} is to {2:one-to-one} as {2:surjective} is to {1:onto}.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464126897-->
+END%%
+
+%%ANKI
+Basic
+What three conditions hold iff $F$ maps $A$ onto $B$?
+Back: $F$ is a function, $\mathop{\text{dom}}F = A$, and $\mathop{\text{ran}}F = B$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443408-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ map $A$ into $B$. Does $F$ map $A$ onto $B$?
+Back: Not necessarily.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443412-->
+END%%
+
+%%ANKI
+Basic
+Let $F$ map $A$ onto $B$. Does $F$ map $A$ into $B$?
+Back: Yes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443415-->
+END%%
+
+%%ANKI
+Cloze
+Let $F$ be a function. Then $F$ maps {$\mathop{\text{dom}}F$} onto {$\mathop{\text{ran}}F$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443419-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict a surjection?
+![[function-bijective.png]]
+Back: Yes.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870552-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict an onto function?
+![[function-injective.png]]
+Back: No.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870558-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict an onto function?
+![[function-surjective.png]]
+Back: Yes.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870565-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't the following a surjection?
+![[function-general.png]]
+Back: No element of $X$ maps to $b \in Y$.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870573-->
+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.
+
+%%ANKI
+Basic
+What does it mean for a function to be bijective?
+Back: It is both injective and surjective.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464728903-->
+END%%
+
+%%ANKI
+Basic
+Each element of a bijection's codomain is mapped to by how many elements of the domain?
+Back: Exactly one.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464728907-->
+END%%
+
+%%ANKI
+Cloze
+{1:Injective} is to {2:one-to-one} as {2:bijective} is to {1:one-to-one correspondence}.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718464728899-->
+END%%
+
+%%ANKI
+Cloze
+{1:Surjective} is to {2:onto} as {2:bijective} is to {1:one-to-one correspondence}.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870579-->
+END%%
+
+%%ANKI
+Basic
+Does the following depict a bijection?
+![[function-bijective.png]]
+Back: Yes.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870585-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't the following a one-to-one correspondence?
+![[function-injective.png]]
+Back: The function does not map onto $Y$.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870592-->
+END%%
+
+%%ANKI
+Basic
+*Why* isn't the following a one-to-one correspondence?
+![[function-surjective.png]]
+Back: The function is not one-to-one.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870599-->
+END%%
+
+%%ANKI
+Basic
+What distinguishes a one-to-one function from a one-to-one correspondence?
+Back: The former is not necessarily surjective.
+Reference: “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+<!--ID: 1718465870605-->
+END%%
+
+## Bibliography
+
+* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
+* Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
\ No newline at end of file
diff --git a/notes/set/images/function-bijective.png b/notes/set/images/function-bijective.png
new file mode 100644
index 0000000000000000000000000000000000000000..5b0349edde748c40a1ce7d10cd32136b3a8e032a
GIT binary patch
literal 3379
zcmZ`+X*kqx+y0pt8c7HlvNK~0jh#uwWM9S%Gt(G^!AKfmCLz_>m+XwkM4s$pLTa+)
z--1ZCv{<4nLkrTsP!cWg{Ga1}KfE8F5BGH*=XKrZdE6iF>pJc`6Nhz_f<Pev07yAI
z*?9`F<ew!W`k#0=ObDW(w(hn7a4S#pXP}sns|PuGx&uJGE&#An0AO2~Vl4r{ITQe_
z1ONcK6aW+>%RBMb!UBlu>Szc2NzWU8C}iSHXLoz?_q$}}wf0G5-3kGKJ%^p`YzeW`
zpRQeC?jJwce{4#sOkAY!i;I*K$f?Iik0fSmIh{ldC)(kk;ze73d*wA{Q7DxZ{GP&O
z5!*4+?)^7vx%a4eq`I^;R|3|jsV$nk52S757PC?MThI~_6B_#W@akg3?YW=)!;JIm
zoX>CWGCs`i{H3^%rQh7~e<6+iG=YbY&8h4XHOD{o)q9e*jxdOY2Lr&5{esGjH%nA^
zjKCi7+{zZn`VHE{-z@&x^nnDkDE1~NM22E<yo9+-Ey7DFR>dmJ9n6g=p9fyX`gBqk
z;`h&r23WNZfX?Yd*vyg^f`Nm8^0?)sYK?YoLwnVk6&HE&AK91I)#cY5b2rTNWT($b
zLq-RZ8Oomli|Q5|4Bw#&bzGs;ck`LOft}9*ISc&`-!nlq+g8ag4=RIv%!k4j+l%zH
z(@oCo3P2<)^^&hn>E{snr5M$j<^B(*z5aJKc`5r2)>_OsFIp2-*h_D;t5*h?vx_QO
z-#@E_cjT$k3kI=IJ5!EDDefI4_aDuDZey%yRIL1@mct$v;Pz)1<|B$G!`@mSLJ@*K
zquxiUO5ZG0t?#@FftKDZYr4<^SiH+VJxB0(_T<k2jr{d`m70S7*ZM?|B#vwg>;8I;
zoQSc_sOU7ZXLk+3_j9&kyN4YE^>Skz>}4RM<gFP7e1F>^UOFyzcHyifuH)e^oL^3v
zRLPvq!30)%)Td;`m7-NeLz~vIR8MS52kj6{KAh{dGv64l{_WoD&4#+FFOd?t8*s?^
zHaZW9y=A5??G1ZvrBvl6+<6l}s$;u3642BT^lp(BMs6rHYx5hz{vOgK@r$OkQ}a%%
z=M2AmQ{*J_nfs|`fnp<<{K)B>!bP(-L%~@fc*82Z$DuDCTcpJNTtlLrgY|!jRVrUd
zMtRcTXG}KF&Nk0U(z@m0(6jj%LMC{ppK&BFit=UuM4<=R;O#~VmS-Fhtkq9vpOAYs
z;%Xc56&aK7C>544qM*C|X|6BMVeRhRgo*X2_2ae7gZ$q%I`%!VHSqmYB7>Z!B1ERg
z61S|!sZS*o5ixQxE_e-W*7Y4Ji4vPHosOHN9NIv6ESJ^Z6x{j3jmUDY8$z-;a9;&o
z-iURR7fmFg|9Ieas#xpQ6Wz_;*BS&(D7^dFo9$}vDY)(9f~l&NRM-=_{$}qlvZ{uA
zm5Gu3a^wg9!0QX%(B)eGT;Hz7nj}YL_SVPaa*Q5KD(y(PaxIdo^s)4Yw;bR3-Mo{8
z>RNYeLA2zzvsMjTR*<5e3^TFRHF`=G6&rT%KnixCd@T;sXG(;vknLhyQy9AGeS^+@
zpWwa+X^r8yj#eog-i_TPaH6_v!G46VdRaBRm+HCM&NYu(kM=a$o9d&<I(HaS9aNqL
zTG5g%c&5~z8G*EFc=bqEv)bd8{9(dg_t12{4u<S=K;so#M}%GCT^)7XOvs(zx>Od+
z;=(v)X%2E0=||}G_{gvK$_ZzJ4i~&M?4EGm%~gT6a%5IB)F)yL6}jV1w(_S)s(J}B
zwvG2|Aa-<TA)koMIF;Rd^4sMVR|vv@Z=GPk?7Gxg&pPMUmP&NddL%k5Tl-?d?!mmi
z!mUF0x9QngQaw|@JavKOGm|&IuU(R8sMsU>o!l6+e1G5qsR6{8*!MuTce_w_9zIa_
zy4UBU{s{gfso_#LbO!Z2C`n`xO89-PaUhuSY^>ouZdB&m1~QtnH&|>r{{_j>kOp=~
z>!I`Aq2D0~4Xz{{U7)Nah9q<^kS^>ncA2E+Zb+S<@2L(Ysma1kLM9B5H79>19E`I(
zWyIn0qVk>D*sNx49DSt}l0O)HJ7E~=Dx*|Bwmh>&RKmu8OmB8GowV-WHz8QR<Z853
zI6h$p_jTlQ&Ak>{wO;m@N`{awUroVCnsQY{EaY{gNoQ#$eXtu7Ug6!1j{;C@nMyp_
zh)poHVnSlzXBs8Z*t2NxkNI$|ZES+8^vM8lD19;v5v5!jk~YZIISu%|3d%tksozwQ
zH)PaBrBnJluUvkGx{-QT3OzD{3;PI8o2TyH{g$D&a~#w6&>+{ZrT*ZVY%#<i;~UEL
zRjhL^*8*Ur+Lh6w)-~11<HYk1wF!w<H_#CBc$$?fgf~H}&6lkWFRk+;7N_xAy_ap=
z_qRM{Vb77Ei1fKK4LryDkrX;8sHXBXy+o4o5bY9JVoFG)-K|b}_rmNy20Hf3(0C?8
zm|lDRY@J>zVd}69jgja)0TsHNEo0I>4E+w*2u~2ZoCP8kM#vaSu^-t@$;^f?wUMjr
z^Ia;+-xU~##x1l5Iz>DJ?_1r;sPTS*(*fT@(<Iay)$EInS<<oNx}>_lZ``)xW!h5w
zyy8ku67t=SN<&7TD%h}>&P#pVb5)Qqo{4c;dZaPiHg;p9x-*(&5z48&I$$|+eet|i
z&E|n@9j_;i0I!#2ULMgy_6Oloiwt?*lh3Z5PYk~8#$g|Cih#!(Sb-{66sO^|!d@-F
zZJ`Cab+~Z%EJxYSu?}oX+bJW`7oeb;sAC_wxQ7PXdE<BTtCex8+#zT44Bi8bWgS|)
z<BHW-{ve{wM1h4Tlt$RBuFM?`hlScAF4I_}+~Q?st(c{{rJ*1xCW`W1Q4NyEWR1ig
zxCt)yc;C|g)9P$z<i01~G=;vcia9~RF(NPUFM|^AnLXg@(n?-H=VaCRY?B$Y-Q%6a
z)`|P>4$*Bmgz>7YBQ`#+Um$8?Q@&z*llWlTLHhP5&mUUFMcTfW$82=-y8=e0<a*gT
zLjl^!N{CP3rI~x9FIO5_zwXXJ*?;I{@Y`rNiH~qsJy;oaV#-)wu{uzV#!Xtr()fkZ
zzF!ryhK=eprBfSZn)T+cR9Q{sG6}!i1u_)P2Y=%YN5K5ku9$Zc9gF(OPucQ@ewb8S
zSFZ4)jh0@N@$Ws|(KS!e<AO$}hq^M%GqjCtx{!L@JHsnoT@q9&4ZL~`=AWUFajHZO
zjInYB@oe~eJ=Fpm(TQNWe`Kd=9(I$zpd%59YgL}Ou=`Q;K<x&ZB$q|w?oFNDLZfLH
z@$Qb09jJX(z+1j3fkM0J;?+y~1wAfzIMs8|S#0%vwM5`AM-{|2VY<mq%>=E<U@!$P
zM-{LJF!>;y=IyNIYjE9)sAx~Y+nQAGm&4&=Ft<BH!T~J(Qd1{QY0%m1rrZv}4I`7V
znJv5b$+v&~(5tZHMkk%|c$;FIl21gS@%{RsWps-({IQFxt5s-pt-Fk{&0oB^q&4~+
zxo?DC0U={^y0-7<ePQK!XX=6@wHc3ltivA@%AziOTAAlZKG)a2m0Z7y6FVd_8;r0}
z>Xq5->pS}I&&p5gr44)$En6PaFO&k~OIG)Ewv-j2_0krp@>65;3n(8>6-oSb6u-EB
zkAlksHuD^j;T0X(+cHVpt>^opi2s{P#ebI_L^h6xxV0)-Sn!krck#!XOb$S5BBuk)
zUBU@nCZ;cZLjTPCgits-%_>(->owawh*gw#e^d;rGBjhBjxINP737qGX8x75sPe%y
zCn0?MRcb5}do9q8U48cx#@p;<`&JuP4Edo^fG8qGlfWO`p5jiI@LFkcWi{6b3=cB*
zfWs9*RVHaT-grrLv$AS^JlN$&&i0Q<m4M6n4QZ=*5w&&tlYWM-m-B9VURD&`T7lZw
zj~~;Z(XT+rLrO)}BH!2pdLto8(;)dDrpCuo$dX}HoN_Iw<hjksjO$>5Li~qpA|l>%
zq+WwVc{70JA@`884<as+rv%?MET783G1Y%lIgl;boTNLFTi<P38hbw54-p@o<mu{_
z>4!s$Zy7Q<epEz&snnS}z7Ce&SNbm<(RJXmcJNI9z;FKL@^#NIamSKJTfh7jXO4xF
z8l0~0OYPG=!SQGH=m+|SoWvs36BY4Wf6e!DpK*wA1yWA9BmuB?&=ObD%U=&w=?Q8n
z4}M(-YOdO2k@|Jac|%TNL4Z2(>_G#ISu$Ult7uj>WLV2WTO-?u;)S`0ZZO5a;(bJU
zF~8y>!P5hwXq4^$BAu4S35s42JafV#fKHQ~@pAjM-{ZHhUO4U#D@jCX*u_o-KQBP%
zdSzM{79sxk(M~1*5f21WF5fw#@kjWx&|%t>nZbd~5Oh#<h!6l{l(De^%EZ9Pl3;9v
tHZemRnd_raXcUS=+fx010{YqDFk0My7c3pNsS*kRXM3z&GbSMEe*o6O0&xHU

literal 0
HcmV?d00001

diff --git a/notes/set/images/function-general.png b/notes/set/images/function-general.png
new file mode 100644
index 0000000000000000000000000000000000000000..7e882ef81be3f55c1e5ab535eb0e466d945a69cc
GIT binary patch
literal 4020
zcmV;l4@>ZgP)<h;3K|Lk000e1NJLTq005Q%005Q<1ONa4mI%3N00004XF*Lt006O%
z3;baP0000WV@Og>004R>004l5008;`004mK004C`008P>0026e000+ooVrmw00002
zVoOIv0RM-N%)bBt00(qQO+^Ri1PB5rBKv?SS^xkJyh%hsRCwC$oq2o|MHa`uxx-CJ
zxIsu*5QqfbtgCn+YE;lg1&Jte5nau$$46FIjJS#*qEUmO;z7KTLx|$RC4z_qP*{(3
zffz+|K@?%a5eS(pQ~O88m`rB6yL!5NB+f7YWTvaCYkpO)>b+N0uON&t!U!XbFv18U
zj4;A50Yk{ggOj-o0AL{Va59hNFg5=1EQ0GBKHWiHn3BL~M>#f<0O-n2+J-4n!YBl<
z1Mmh{h3QbkLIf9bE~~>-s3Vd6>}EUt!c=I%^#~p@s<6lwPS!4<BurKa5GG3kQ7!t{
z-Osh0OlSJ>CH8P*TZ)Jl{o8uenYJ8bKYPMW9SB|d8}D%x(eQVace#-+;hxnL{h7uD
zA}M7RJNOYh@$_H_myt{zi<!X>p{ra+W@AI}5<`GXSWg*`hNv!`co`SRc#I)_o~H3Q
z$8eHQ$56KBI0R1s(2DI04N)!5p#nRz6@{$}b8t|`S)rusP3jm1yvW}}RD*kQ@g0BE
zkTRX!xVR^DWF_%4#oWNs5Y2mY5G<md#`Nu2gdi_;BlcJXm9!5DRnH)pC-u`Kn2lh5
zs4SQ`e)Ni>)tHB1rjC4%Aeb8(vgYvs`}met7Qgov1i3o%<{`M*daACX0Js9d9E)c_
z2Wxa&xRI>D$p~wyO6ChX0PqGL&b92l-8jHb&BMAsbfAczgVpS15X)x%<o!Q^O1`I!
zMQ?tW8t=-Wehi|Px2&S-YBo^LRPU#o_zFQG6D@f2bqFRKP|n>5u4&oEayP0dG-T?J
zVk-x%6vZ|SWEdw~CV=n~we&ZrtbsU~Z$+g~C6}T`&X8T?l4kl_2jSvz1IwF(n^beF
zoXT=M*jdXIvdAEddsvGD4+RV|_06j}NGAi!OQ4DsW*QlH<KPevc$J36PMc{Q#?GCl
z*<R-(xLe`*t31gI%$6#13c@Unt1YV#EF&Sn^F-c8u)?G<Y(0C4S9snXh@_5Ki8ABa
z%ZFxEwgX#msXAkB#YLeBru9%fiv6k+OPR}bgyhbd90aEsKlA+uJ6CElf*gEmxV5;5
zBh5s=F>d1(0BN$?bC?B2k5zBs<}!_`ufWX`L&vUOI92Assl23%EO-(pCmBiBGz9;Y
z95|;M`2L4+(MvH!3dIy~9|y^jEV~bGW*Ar1GjQ{oWS)h%NH^&Hk(8S`#q$AWMmYrI
z*h!hRmEWFniWFxr?YIcRI1|ab7Qy*OiGYU-13tZpl>wfN#KVJz)L<c1#F<D|Jcpzu
zC&18pJm72*O&$eo4w!c_$LJAs4cbw|vnCr-FL2a|&AP8rt2kGTQy{A^cJCY&3^Ea%
zV=`IiAs8FHW$xxN#T37ib)Fy1Jr6f02Q7sAP2VIH#Q|Oowvd;p@$M>DBP)jKtYrgv
zTux#m1Lzp}LH-PzSKTxh3$O*(BweZHX_Yih#W0$5CbNv~EFha?;CbwH3!)Aq5R9>a
ztV{%F2UEZ-+@wg6)mTZ)W&vARjfeaowD~-0EjcIA9*6G5H6V(;ysw%{X@Cdl!IxCC
zhWE(f94U`LJ-%Q))?g#s^k*KaIzz;g&0;FJQ*jT?cmx@AVKljXzy@;3&}?S6$IUd0
z$a)Bu{sp!r2C0Iv*m$GiQ(H2a&KfrJ6q%}i&r!zd)|%{ui}Yn}MGf-|s>bsiZD#mH
zlTHrrQpkKJ2Gryp!AWb2$ZCU=hxBE=9HC(3$l($Mmj)aeQ^{rl8(Bbhv*m-$Y{fck
zW23%|FLGGNa2!Ywt9f2ERSFZC&wAb`hjb!=D2_||)PutuYV_z@5&Xy!1FKQM9?j}Q
zcQSd3k6FvJ2qsxb)+B_UjRB`13|)-B5y3!d19}#MkNJpPMk~6MjlyXNGJU^PcXSO0
zHW^rrLT{NeI7$>wE@3n)8O&>JWj>iI##i%SIvJtat5$?d&r<Oc_8Q!3m5tN~tysfX
zil&q{UzLz>LGI9?hBw0=78^{~Qp$DI;&nw+Y8jJBZ!#7}aJ`S*#7L2~h*FyAILhco
zuV7Svzxa%+>C6F%9P#vIx8Elv<dYBd;A|!`6Bk?A!gid@;IG7ZeP|OH!*{r)YWY3m
ziNz^v=nOZQtP2oiNaZ*gp>d_=S5s;;f?9e2sqCYTv=~l8$eyjv!4q(aUM~_6{7c$4
zm2sMM;c~u2AlL-l?j3#P$U3iild6f~P%1cNFj-YpO3pNWq4@~XOx^8II)h22J3q0D
zYTyWCffzhGYFPrlQ%AI_02G;^#S}?uTFKbi7r^aPPf!YV6jQ`XW>N#d-3%cU8?cT0
z=*VH|IolyiRx+QOKvt=w$Fnzk8ug9@QW;DtLukXd?4pPQKPT+-_?`^_Y-Kj<D5eMp
zqbbn%P7iy1jrwR6A5sEkrjS*C!c;3dbAZ$)LRLgR#ndV0dbi=fT(9R-?r7fuU$r+{
zrC+I!73drf`m$3s-7i8m3_Eu;5i%=K<-1~_Ox8)DLxEIBInXBnS^k7<1nWa^G_T6n
zO%$!t{b9ce)z}YBy14I5<||}L*svZpIWbHEs!VA!NK%-|nxYl0OsR!>E!z^`I37f_
ziu!0bc#QZ($UNM|(10^pQ*;FBiQfnu0y=0XtCK>D_Tw3f7{u!=lT;Ur$S0GPnn$5{
z2uZvT2Q*!#Q6@`idT+;5tYt7?u-b1bWg)AP*vA|Tat3W^$WpSp|Lb*~#M6snDRr^F
zB5_h=IYLBMtPELdmF!C`<($PhQfb3JcCv$cl&GkS;E-4Xi3?HBavImO>M@3=spTMx
zDdM<PMdDhT7{;hL%ueA9#z2D@%8;eKXGH_95cRB>#$PBeShy6)?9Ftp)!F7UK1jE3
zr7c3NZzDzOZvQSk#!Xb~tVo>1knQvxXe~mV@5B<Rdt|wR@z`+V=4z9BR<p6g!k*<Y
zre{4y6wyQzt*Vrs$UV!covdR)g#l1jcWYXj7*;jfKx-r-NR;p+*u5stZ=S3;Q+9E-
zCq}}La0sMUm~NL`Aq;F8`xx+%DjppqcNAztlDBNFJ*!5lhc7g!mWfo$^eh|o4#3co
z1yzzw<}s+2KD}R=tO_vpuwsKnDd(=XNspZmvyAQhBgmd*Yi>#eMk$zb??bXASyj^8
zi|z+HQ_Qv|yKhYGS=AggiLCt+3t?c>H5eRVJ7-!ahEQq}S#k?(74+71zBGnf-F1wj
zmv8S^t3t6UK}SyE8>#eV#OXNo(~&g2dyavg#QHX=+VcI^WRYm_+Da1L`JYsd`mysz
z&9>w^W)V*^*+HzQ`&nRCCkuuI(LVk#)LX7h&w^yOXiQzpD1#=UQaLgF+WX^-L>jO|
zDn~gU%Wq>!rp`Ckw;EYLvnQw-#UIY3Txydf*r_xuR+d)ur^44it3JnUHay_rIXbqA
z@31^#^U&twsybPP^fXii2GD~->0Dnk$Wk{=90T-_wFwN}f_pCLA}hw^@MXTrO?2}~
zNS&-Aellcs`30lc!?!xf!csv@13f)Uu!2#Bi9j3|vjQQVZ#T6TlU1YdMuNXc?^{>u
zSAvncau!;8-f+8K-LjR2peFrT!)#WNt&;8qKPOqWVU~p_S=-{$dy*9vtDccQl}o6P
zKc#fQEuMqAW1o#gaDhc+U9A6%bp2u7v6}jzP;?fK58?oEJ6r}uN>aznU{2phu}8y!
zvbNmm4GJ}yG`}xfLwOLPyguh6i^%$f&-{HIttoRIuW}##6z7N5Jj4Lr1ETl?7m-Xm
zU>~KFQbH*eBy%w{gYb`U;0~$X^Lo*NGlM&i-Gy2W;R+%t;)*8UB#=%fIm~Ar`>Er7
za>*ux6dgV6T!bZet6qcff4zkE$S^N5fj+8hGH>y9fcqpmQ_eziNudYbNueik2udlT
zgc82zh=##nAFC`lR=$V~x(ByHGzhms$jfXx0TFx?FevA2Kk>8TNg;!5a#_YYK4uxY
zOeT}`06|w4VAD8o=tmSbUiSY;Cg#v9WHBhf9??<UNEtoZ#jF6{p>J5As)Z$x((spn
zn+k8gE1?VzFpdI->#!_kFoKU6%R9l6)syddhwF4zVJVl<pP!`i#L(MEX!mj*8=0y}
zXlc6PUo+-2fo?|Z{-1$Im$G&#f(L?rdps$u=Qw#9g?71)m>=;R3LO1SF7#I%pOe5T
zlII;du#<zFZpbFu4AwE;yCY0lXcMvVu(ydaxsp=)u!=WT%Add@M(~L-5`8)@X@g}k
zPE8k{X+b3AtO_8s{m3AjM|qQtY-AGx4`YBqR1yz#&~PYyEu}_Ag8sMUaqxQ)hTix#
z6BoTTq??GKnvuY*0fWu@qAzY^O{xf%;^uOVsmD^sLPMi=_ryV-hIBc6;f+#w3m<4K
zdoE6uAlK4p&8Ijt>C%}v*=+c(y*H@VaQEf#20uwEhcr=;z|r6%lr~3u3UP6(N~+s%
zvC)K`gK2mq1v`svEaw@OFnCiD=6GY<umZu`0ryQM@eYFbOmf$KsB(veKE<SxwPtC$
z5!{7?!#vnbBodg;5$xPy+U+PK5KJ++%zI1-B;#W*7U03bhvbmOMPzX=>u^%X(xBrg
zD`O>x4BbiHiK<4S4NOBCkFvATjZQmw#P7w##$q5YLt_9wjhn$(gw_n;Oa@rCqyR5s
zH>~###-Sl>;|Ywe*w|u7!1E|RW4|f8vVTLkRywp)U>hTo7LT_%E*+WD7pd4obo;gC
zqWCjKL51S%NHIUPbP@aoXJh9B-6K~ND{-2uUH!IjGeVPL?cZ<~+`#V&a}kn}vFh=7
z5Hz2#h+r;)=cJ6Q2<9Mo-l(gtPAs~fK(IvfU5f2_6Jh9`t|u6#;9_^f<=;y9Ge6)|
zyiW3WfMJy4U@l!1(sbu(oRn||zk77#1)S9IgkLA#spL_EljrnmW)UrpuoNC7izv!i
z%?>K5r8PZB<5K!i$3kY967ul`MrUqdIn|9EM@LxBq+m7mh!#y+PrA{D8un2c?m=OM
z5k?qcgkc2!4~1b(V9qa(iU0rrC3HntbYx+4WjbSWWnpw>05UK#Gc7PQEigD#Ffuwb
zIXW{mD=;uRFfh>cf^Pr-03~!qSaf7zbY(hiZ)9m^c>ppnGBYhOG%YYVR4_6+GdVgl
aG%GMLIxsMysXEgD0000<MNUMnLSTXhF+nW=

literal 0
HcmV?d00001

diff --git a/notes/set/images/function-injective.png b/notes/set/images/function-injective.png
new file mode 100644
index 0000000000000000000000000000000000000000..425e3398cf7e3c991289b6ac80aff018f310a617
GIT binary patch
literal 3276
zcmZ`+XE@sr*Z##GrKlJYyH<*(wp7(DQPD`E4N^6N+FQe)KB6S8QKMAN4l5#7S}J|e
zDzR!6vDGN0s;%lkm)G-N&zJYZ`{CU8b)R$HpU%0?b<QoclbwhVL<j%?5v08h=8*6H
zCj@x@7g+&^#N%&?vIKypEaBf?yoXxJ#~y<MfOvHPNJ;^KpNFZW4FHIN1Hh&y03hxG
zfOL4tGiS5I0+;uBI~(ADnl<?RQ1M42QMUZ+JOW%AD!-C6-2gy%0%>E3Bh7v+4Itq-
zvc3N*yoJO|C<WJer7Q7>HClW~4z}e@pA^AqUI3-j{yg}Im44E)LJ_}F0m)m!q*^Ql
zHi;dVl9mD`^6&-_fXALHDUHYO3?4WAvjq3^3;krY`^L0z@T{U<|C!yv&mFt-QHzh;
z`MuM$s_y+4YS^`^=`^ctF6*RQ!4Fb;RR;}Hd<|BVseb`tZaS5qM}$vH@?78=rCg9v
zXQK$oo>d18r~B?!C$u|oOM^zDYLDCW=V5H6foG4YKXznlg{gaKGi~b5yb-*~mjHa{
z_ejp=DP>I!TUb>B&oP-H#0>1G4J;w+k=drB+CTd#9ZcC11<xb18v<0p97i5uqCV$)
z8;FqU*@PpgiVx<z!pUwuxX}O$Ax014hfa`nGQK5#vN8R2Rl2k%l3Lm&vN}rj2?lxc
z-H`2;flRBBYZN>o+#3#8Vli?-!KQSR^vL(fOS4f$4x;waA-T>m>$`13S}6zXlC`YD
z(AV~0r+!4i^HcKDY{_k2@I-F+%n>g=S%bzj>a-%r8fl+DkWMlkI(Ed_1d*3o($l4f
z=%ML8%y5&ck@^s-U3FVRH*^8_b$dsoE2aY4>CrYn?pCvc&+t_1M~rJcmauHG4O@`1
z1^4O;M2aLLq3L3)%YnYQRTWZqcbu@;>Jwf1oH()2%vi2dr;fTQu}SKUb{}Iy%rb)H
zaRE@jtWQEXq`c-zHoL671jneFHRkj8;I3I7+NdrfFF6VCu!K8NuTl9u;zsT}xiOFP
zyQIMJrZproWSEl^DjH-sPQ=)i)!~pGi^ryi3$&MYTJ-PR_|y1Sd>{JJNzAyeDR+83
zdqRKu9DS|i?XM?!1t5EWjavDK-tu-xPJU#Xu20O0Ku~bS&m_!!*07AQuEo_SkQirn
z5L9slJU`@oqae56EWeyh=#LkrxLiT4V#@;G{d;`62e@s#zt~&y&O4UAO3Kid(z|`M
z>(t~GO^vB-b{_o7+Eo?BF(<>+yE~?RM>Zw1M3gcMxEy)W%X0J}kCAYV-CDWxm%Fa6
z>MqoLKSKd_RrXjYoZ_}8x|DbO%+onXwu95vH{$v|4$CI5LLL>Z9(=pgG$xMl%S^T9
zaA)1#eIV=8i6w3L>T_g$SgxJVL#kGs+rEt!8j)FyM;T||m+Vdz$15=N1G(Hay#+_G
zFWz4N{7M}rT`ASJQaly!;JCLKiHVR+)-v)&S;Ng8XnpH9&#5Y#sYNjQ=jgUrHSCbY
z76;=bTOxHr1B?|Xc<WE%d8TYZ;C-AMTpn2)v<g4Cx{nK`lLUV?m%zBCPilbQiV;2l
zlQ^DoHxuy8gb7ncITgX(g^$EE-4HP;Kf2$#hGt3XoX$XrdWCqm4nNcuq~&E83K<v6
zznpb6liqaD0_Ep6oqKDz9iCEMM~WFPF1VAX6U;VEWIVox;er(;8|#;Zh<b;tNh8`I
zS~ROn!z81GcS@7Gvzz#4HC`856NOw}s~WrCfxV1DYstRqnxg1`euZFaE%9~vOD+4-
zM15qdi@N={`~+*KEvj^IZ9eqY`y)1<*;<Ofg#Y>|a}rS(&EEZ@eDPQgIYF~JKN+`W
z<i!}#;t|cR`1jObjgJ>gnnUkyMqZ?9i`BuKpsr*XDckT?^pB$JW53YWTG^9C3kUa)
z8*J#N3#P;t@v*S-#8U&kxGcemy@T<r{`sIiw;vIzf~1KIPO(WEO74SOStwV{C6`wF
zN{pJV{|ZenuX_8Bg0*g`2_q`Hx+X1R9Gm!l;JDAiG0Q9mkf8doT>27*@#trg)$@Si
z3D-Zgspb-k8w21SB`+rNJJEccE4|JsQ5ZtLkbwqA#`IYYCYFpkpA5wCo~6i66csIw
zWoq2(LSp@NtDU;==V*EHkA9Am<(IK+YF-w%K~d|{%>2gJrmC;cI-}(FoqR^;p&a6{
zD!H;dr+;XxsU+<Ha<9R&w56#-4b3~{df7w*#8B-3P7fg^V3+#NBlPg4E?MWTz-~;s
zN|H;ZE3@K(kikW{S(^=omP-+Gl@AY<oj_sc`;^<{0fKn|jbN!`ts>!CE{c2CV{Z+<
z%HV;o-Ixo~4mj+ui?lL>$HfO}d6@1NSg_6t?9ZjZe#jJEKk^n7mhx%Ybx0wf0vpAD
zr;q;7vgAO=&@8LqLNO4-h65u&U<Kr7U2p!u`!ml)WjmDRuH`U^ao$-E)v~;F?#b{s
z&Edz;Y^s6$>B525lPt;C_YJl6BVy@+*cmeycZEuR-{U#`bre-*Dw@r|&=r^&+WIcf
zQ8e|N-`R$Um)kR?rZsNNK}>u9^~JsyG8hrL6DctBilf!(T8t48tsBMfv}CnMHT`a1
zJ2wsG*sK36H@sJv39$mtW8Hjr6u6$O8wqK0Q@)5_gWP0Hxt428KsoHZR8D}5)f$sV
zv~*YSOAc!p@Xx4D2DjBmL1)Tx^9SBH!bn42xh`Ig!XRl|!ouq3<<+JfR#DU4-8*62
z9K|jn>H`ZffjO`tdoH2iTpCwZ(#O?{^;|;z*jv5JRs5Co{i5<&E8{WLm2@NUQ)QDm
zo5?67@5)|oe_F3yW65qIeOl>LrL60mHYf1zz6$7v54|%AOiNy*ur2t&lqS(^JHhv>
z*axHp{^zQhg~8scBN(Z0ZSJ}NMZcO*Z}VDBW%6~3^+Evl6wrA@uH9R@fLU~nMMrM+
zx(fWOv9u4Zg(SmT)io<$PbOwTW?mTnCOao|x0_RyZt!K&W@E8h1q?Czo>&-*aEZa#
zev%b;d`Hfsit;>4*B5f;;V_!lp;$;WJi22)e24Zf3bp>|#g|gyztWJX6W>zX(#fK+
zH@8+rCH9=USm8xN67L)QuYb#YmOPKI78j_W3fEP!JDkdJ$so<fxpji$I@Bnq68=^f
zgDWcga8x@|Ea+4<cW7a5lz1H7uX*Nme&T)2dgCoSDfd6asoFT9_#;O{<dXxl3aG_(
z#A;^GVwAG5OSZc5UcNQHK(*CzoHgA?sLx3oXdt%zZH<flAl`<!xL6G@2r-yec+xAA
zf8qX4nKfQ+rv7w@_+qVzv?y9@YVWiDK+ZH&zBPL{rrP7YMrgn!>!u{QJfEN^7k%vI
z3y)%dT3%_oz4P+B##^+2p5|O@zIY4n8mrRuKJ*tKd<Cf(wN{lTEl7opvd|4~TjkzD
zFMr6xGpQs@m5_o||IXR6FVHbOn8*w6$9*js>KS;O*LlXMxZ1%>*S*gN-cIw+`2745
zGl}m;ZA&H=biyR`!*DwisxR)3Dh=d$rDeG%{pF0E%3SMWEKJw--p#QGSH{IN4mUs9
zfv33wkniL2`;z$)*Qy!fSJ49jV}e&W4Yr1vLd*@p7f=ciViKstRcQ*lKfXAwj{W}d
z4e^dcDeQ6oT%;*$HTRX#LYVPaTm{yy1=Taq>7zoRh||sgJ_~I71e-GhHzwa{+0vW2
zh2>MtgH*>Skx^#Vg)P`qdFR}XP;Dm}3NdGJ#^~;lj*+Qt@ITRQj8DQ(?;l+5c${GF
zFR>;mk%-H%4u8tV?<)Nc^)YY;1)2H<_!#D^7P<2^<6wGU+z|CU2H%gQVPH#ffwG?B
z6H1%ffl&iU+iUc_zTDB3&=xFmO0#OEw^aUbu%-D|F^d0EvjJ@B%5P7TKF>xz?5iM~
zv)~0&n;h~&sTC)uVeieem%69G@HDr{;(FD22K!it;<yzp$z7~UZqii1Z#<rsP<S?U
zr7hKqfIeTTavRDs0-5){ekyrAc;45;38r*gL;N@PTK0sbhpo))E8Tb+&?<NGAK4n!
zwUaYhy)R;OU)%8#Y)-|jj$T$98X-ZBE=GdtiiI}%FlM-~72elo_KDa`s(;rGxuRUi
zKag!|))y9meTnFJPy(5aG_u||Y!6VnPkkw11`WXh(Y{7u5faM>rk(DC%r_c$$pduc
z>Gu_{ZwHEEZky%af&KTQc5D25w2tKB`g~FOKR<+8v@JfG=oRgY@QL(2BtQ?Yr>_Gy
z(1Dxa;CcvsQ-r?32{;@9hmU?s4*NfbkT7C^U)=w1cvyoNKQsVHTPK@(E6?Qr0s4OD
A8vp<R

literal 0
HcmV?d00001

diff --git a/notes/set/images/function-surjective.png b/notes/set/images/function-surjective.png
new file mode 100644
index 0000000000000000000000000000000000000000..3d6b501f359b3656d6e1781bdae8e5a1a87f69b0
GIT binary patch
literal 3332
zcmZ`+S5%V=v;E*uBcUcVDS}cAEhN;4g%(I4)C2*MUL;711ZfA6DqSMM(1ZwqBoxtr
zMnt+ORiugafIv{hBOo9L%lY~5x-a+PKFsVjvuCe)m|3&d<hnT7fkBEO006)kdmFa{
zEc>sC2>g$K7=8c(VU{>c0JvLx=x2c7L3||0-VFx;NtytVmI(m+2dT7=06>NVz?we*
zplARf7gN>gYH>jD2Rhi<0RQF`kM$fx!f_a!t?;UV2)~y4k2Gy>01y+y*jVBdUVo_$
zPk`8}<os1GIvkUEhI^Y216Xs<+n$$_l!MF=<OH2n1Z7=&nu{M1XHE(@4<7LXTiLvW
z@*ip%O_vvtQ+1WR`t%Ul3#ulZja1?bL?G23tIB5wCr!PZoqp$z;oX!duWw3z|C>xE
z6UehYM_Rcb@^t>!Rdk`?IUtB{kR-fpr*y-opj>DZM&27fdMNGU*5(c82C;11nMFas
zn^UOf<};Y6{Z@JZ^X27{8$D&Qd827rZ*@dn5e)pKLRwUc!->SPwK2Z#Ake4=>`1HS
zA`Xzf#PJDviPJ8f_JuWALD0x_<9pC;%sTje-WF3v_|>pU(rBp1sVkwU|C;uOu^+i6
zc$hRr@$?qZ5FYl4F(fOT<V77`try6jLZJu8i9Wd(121e`vK#rQ<o@`yuJeo?H_HNH
zHlhh=8m9d&ScbgOXj^lB>*UutS!uw`HNCGy9lc$*U3=(@*|6oVeOwg9mAv$z`GWxV
z<Ens8Irb&b9*Ubl)%3&&N97&=aCp}(<K`uWL2@uDk1-Z}R;++>9Y%dA8beuCfCPN@
z54|LP`p%S^lW`S-NUJMvd4!Nt)ewt#g)f#~5Y{)kzNH>7y--qyT?y0&IoO-fPL*qD
zFMDs#M7rViebH)uA02$gEa|}ILv`w^S&feC-*rKWdiiUqkkInCrgB-y-r}<{DP3Wm
zdY?j#Q_&q_Y$Nf_mPLLeirai2Uw*_c$?YzMy>P<p+nSwmwCNQ#M%%)_FIL5B@jeen
z9SSh<-nzciR&FgDl4Q<GC?rmpUXN1zy4El}WOua%|9;`?i;Ig_#)1UT(JU!Q{4Wvs
zIa31v!!GE$-t3$YQNjP{ta_5*X}yK7K>asC9c6kT*v{U#l$5T<L6K;)CT@S)hflU5
zL;D!os1~#}UKI7_&a$Ek@7e+@_#1O*E--c6d)JQOTeyPsy}0~ndatxwsp)%9OI5JE
z-vy;jZ4PZ*+r%blPW8B!-(E15M!##E3vQovq&HgaB1QcMBmcxK6$*Lqt%JMS5k3ar
zfZ~WsMvL0a=Jmyyp(yJ#B}i(3%xsQ@lt(Dv`ci5o%hQVSH`Fl5Bf04AGfj3Z-8pO=
zYi9^2>cZH6?}06Z$7zBFA66|YSrfFC0K<Rpm*+K#PQLShZPUOhMW`5={e4HPW2~J@
zG^(8MvygQgyhML}6Nit&>Lm=Cwsw@#m{&$IPYBwMZBn1(A*~lR@eq<{V~J#Smr~33
zcH$I%7;h+%+>s`eJ`lK4AE*u*h#czV&#J88gJvnU`<qdGUp`a&J-5xWz;6icMo2;V
zwG6bNbuRK>fjJw9e)I?YvfGzu1(~~h0=_}Rm?x{vPZc8lBc|yjySR7})f1UJhH{E2
zR1H4DF^>Mz0d!8A38mKn%qUHRQM`_dh`d#L5su``SmGTtm#K_(ItSi$!k$?va;bcH
z-C23?_1F_3qOqF`YkbWiqB~A9;zXp_*lS8UHgZlfCN6$(ci}S80mQZs{pTOwy@khK
z*7~+@)w@>3`Nd59%gkoYx|bt(cT0DI-&j3KZn*bDut0FXk{xtq;DoH(Nss8EpnwhI
z_m$1}(y`SXp{)*4PIsVkoA8}Q-yfWP)CVS+gFnl0Gq#Moll$qJ6tmy&UJ@p(@zaTu
z8AR;W_9TP)?b)(#G;|9(&!c|2XgIs53~izcuUMPBvLB;t-oLta^a16X-)QqKuC__!
zT!JrKI2;e<VpeL=c}`rWB&tFB=SfNmiu7#ZP4zx($XenHOJ`}YQ>#CUsxFONzbps}
zYRg-*p37=BcMsj6&rQbhE5jbFq5Fd;*80SvqG^#}qUlG0Q&M#Ar-kfX+G)4&TAvvP
zs0WN>85(ypbe<y%ml`hWKWz*u6Cu>!V(Pl>StPLq2*2m8t8-;;^|$n)=9GWB$NYs#
zok|RkDq2GcQJ(Sp1S6@@dW|1W^DjZ0ryH9iLHNZGk1+A4z~>lvuTu~ubEGB7{;5#Y
z4PHEkIb+_WF60UAG(VU_c4$Wfxu7-RL?N-x2oD}#1K2NUP9L=|6s+9H9#Nh{9Mdc@
zPN_9CK`S2AYNWGJ+{d99Ylg|F+|gzm9H+jPc)`Q1pq1{248vGMd%(EAvPI7vLBDJx
z)SY9{56BkH_CK*R?beF02P30{^8Om(vWrtL4+TfWlkd=<2j}vRP%X?y4MHZio<>_c
zL=7myH&q6sIQ4X_i?Q;F3RFz=)+<Zdro`F7_pAzBh#@<6XW#VAhS9s@0}dwNL3Y9J
zt{R!=2Vm7v+@VrOmCqlC^~>CkcU{&his0!k;g@K`%*pzt2d|(hNH_%_z`^51*{7-|
z8pifPX&DLjbZLWdtK={jO!m+&7Gd<DhLEhkC@r*gn5Zx^@6s-WtaBw)zgi3>(8j!g
zFWemUku<S!(G}wPg0o>BI}D*!{3G+MoQO~Tr?OpT35@}b8^67pNV4sVn0x7Ugl?pn
z{$jy>lFg;!@8H5Xvz$lr&tp+rs2^!~Cp4kAiBxuu%ITz8Y(M{b>u|#L!!5ni^(4dW
zkgmR{Y)+X>-;PmH`6;=KQ0j}Ld3*ZgbvQym*HNX@on3B~56k*=@k5fzs^a_GWh_UY
zFkte68aKTveHEre&pg)G|4)0v4G>qpR)MH~ZwfP`Fgz(gYYS;p<mL&_D<{Rbilpur
zTuxAz7j2V}scQroaPqD??t*GIR6yEl)mC|#-^XpD=W{AXMt9!YnHvbz43v-$BR#RA
zx!Y3W<VI&(KZvHekNd3xY0&+kOjy3xMa`%FnM#LP{g*tBlZ);omkuAVX1)$%TKJ6q
zSrx2SoR!^VZcnXc>)kw@M-5?2G&(kFwP%xrZVV)t#2B2$SP4SRyV1Q@$hXfulppXd
zLb{rcT_4A<32(*ko4;10*pv;~FC(ED)C4bvx&2GNgB#LuyCx$Pe2iYw<<k?mQF5@r
zrG@?H#RGAcu2hPz<`!Q*_Y90J!svBkE1!>xh7-Uj=X<mfdk@3AmKI!3cVl*8(Zu>2
z*@{=Ii%EH&6SQla4}xqNbzvLRLu$-;fdpx@9HskkhaiN5E>rA+5~znH@|k#WkB5>w
zp?@^8S`lSN<!7{`Zav1*uc3IKjpC~s(i=tOC8B>RsLqSWNeQNmNnEtnWKZv3(;Jd`
z-Y_&YuN0;ubbL?1L$&=TJ2h2~9w?B9YoeK~8*;(6{w=s4%>EPcHq4EfXErlNkAv7c
zLJnp}CJiEl3CD6>V^O*k^7z~{=pPc$E~j;-;-h1?@s4K<@AC8Dgr#rWO_>MOrld2E
zIT)<IfkzO=<bGCoG)Kd~A1q5szu(~&c}R>^Od$@5zb)O!FMG)`X63zbxeDTQ+!Oy!
zboHB2SDDzkL6%L;Oej-6Lmk4P%b6F5Rv4G2C!G{3Aq#`f&O2$qF5$Vyp-JwXv^h0S
z!3E_fnsaN?3~8!4+n>>s)7O5P^oIzIYS}V8lrKVMj=7f<75fZ;Gk>kGituIJk=fd)
z6uGH})qs7&_XZXnYC(h>g^`yt{WGU7FbiNPn2@_x`-R)C92F#F>^%%tmsbQzsHxFa
zeg272MK(GOL(@0fo_H45!tW`#@qHy3<-JHFGDx16hie<KwkCW$&ucr<9ox|j_+BgG
z#|SC>l@!^IznAN8ABLs`VlKo=oHLVOS+IS`GJAY)KvmalqE5U2=thMibi*K3RvaPU
z$~V(i#dEeb6>qclCu?<}Hk5xfkFHC*^WAhG@SYCI%(=X2=T{uAj_*CG-B8(<sO?B!
zaS@dGU6uPQ)GpQvM+M0xYImt~M}GQ1cyG8PT~+wBd^wlBoW&!oFJ9T|?Pce`(fJnw
z_}7D(S4(fU9qQ>NQO1iw+=9TA<rlc4<K|tNHSc_|qYeD1HsCa&OY-icb$$}7|GAFq
z?A7`@lI_*Gd@Zhwghun?;Sp8?cojj5_%wVRRIXsVu*gFTo{?*4OJCBa*I+&1ul7-Q
z#qj$18&jA%J;HjUQsOZ!f=}!d=vkjw<A7hPrN_>){RWry4_*b0I9t!S;DERgR1hiT
z000CWfz*Q=>A}tLa0CizjzSvgz~Lx3d~!QI`u_+bql3dk6aRk!orRh{5C9lkC!1C)
H|MdR=g?Q|5

literal 0
HcmV?d00001

diff --git a/notes/set/relations.md b/notes/set/relations.md
index 7a22f75..b91ded6 100644
--- a/notes/set/relations.md
+++ b/notes/set/relations.md
@@ -345,6 +345,24 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718327739961-->
 END%%
 
+A set $A$ is **single-valued** iff for each $x$ in $\mathop{\text{dom}}A$, there is only one $y$ such that $xAy$. A set $A$ is **single-rooted** iff for each $y \in \mathop{\text{ran}}A$, there is only one $x$ such that $xAy$.
+
+%%ANKI
+Basic
+What does it mean for a set $A$ to be "single-valued"?
+Back: For each $x \in \mathop{\text{dom}}A$, there exists a unique $y$ such that $xAy$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443355-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a set $A$ to be "single-rooted"?
+Back: For each $y \in \mathop{\text{ran}}A$, there exists a unique $x$ such that $xRy$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718465870483-->
+END%%
+
 ## n-ary Relations
 
 We define ordered triples as $\langle x, y, z \rangle = \langle \langle x, y \rangle, z \rangle$. We define ordered quadruples as $\langle x_1, x_2, x_3, x_4 \rangle = \langle \langle \langle x_1, x_2 \rangle, x_3 \rangle, x_4 \rangle$. This idea generalizes to $n$-tuples. As a special case, we define the $1$-tuple $\langle x \rangle = x$.
@@ -407,6 +425,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1718329620114-->
 END%%
 
+%%ANKI
+Basic
+What does it mean for a relation to be on some set $A$?
+Back: The components of the relation's members are members of $A$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718427443424-->
+END%%
+
 %%ANKI
 Basic
 A $2$-ary relation on $A$ is a subset of what Cartesian product?