feat: a lot of changes that I forgot to push

.gitmodules

@@ -12,3 +12,6 @@
     path = scripts/precompiled_html_generation/fast-html
     url = https://github.com/simple-web-tools/fast-html.git
     pushurl = [email protected]:simple-web-tools/fast-html.git
+[submodule "scripts/multiplayer_development/fs_utils"]
+	path = scripts/multiplayer_development/fs_utils
+	url = [email protected]:py-garden/fs_utils.git

html/fundamentals/functions.html

@@ -38,7 +38,7 @@
         Suppose that we have a <a class="knowledge-link" href="/fundamentals/functions.html#definition-function">function</a> \( f: X \rightarrow Y \) such that there is a value \( c \in Y \) such that for every \( x \in X \), we have \( f \left( x \right) = c \), then we say that \( f \) is <b>constant</b> with value \( c \)
     </div>
 </div>
-<div class="definition" id="definition-identity-functio">
+<div class="definition" id="definition-identity-function">
     <div class="title">
         Identity Function
     </div>
@@ -309,7 +309,24 @@
         Inverse Function
     </div>
     <div class="content">
-        Suppose that \( f \) is <a class="knowledge-link" href="/fundamentals/functions.html#definition-invertible">invertible</a>, then there is a function \( g \) satifying said properties, then we define the notation \( f ^ { -1 } := g \) and say that \( f ^ { -1 } \) is the inverse function of \( f \)
+        Suppose that \( f \) is <a class="knowledge-link" href="/fundamentals/functions.html#definition-invertible">invertible</a>, then there is a function \( g \) satifying said properties, then we define the notation \( f ^ { -1 } := g \) and say that \( f ^ { -1 } \) is the <b>inverse function</b> of \( f \)
+    </div>
+</div>
+<div class="corollary" id="corollary-the-inverse-of-the-identity-function-is-itself" >
+    <div class="title">The Inverse of the Identity Function Is Itself</div>
+    <div class="content">
+        Suppose that \( X \) is a set, then we have 
+        \[
+          \operatorname{ id } _ X = \left( \operatorname{ id } _ X  \right)  ^ { -1 } 
+        \] 
+    </div>
+
+    <div class="proof">
+        Let \( x \in X \) then note: 
+        \[ 
+            \operatorname{ id } _ X \left(\operatorname{ id } _ X \left( x \right)  \right) = \operatorname{ id } _ X \left( x \right) = x
+        \] 
+        as needed. 
     </div>
 </div>
 <p>Note that this seems quite similar to the <a class="knowledge-link" href="/fundamentals/functions.html#definition-inverse-image-of-a-function">inverse image notation</a>, which looks like \( f ^ { -1 } \left( S \right) \) where \( S \subseteq Y \), note that they differ as the inverse image is a function from <a class="knowledge-link" href="/fundamentals/sets/basics.html#definition-power-set">\( P \left( Y \right) \)</a> to \( P \left( X \right) \), so it is a function that maps sets to sets.</p>
@@ -460,3 +477,12 @@
     </div>
     <div class="proof"></div>
 </div>
+<div class="definition" id="definition-support-of-a-function" >
+    <div class="title">Support of a Function</div>
+    <div class="content">
+        Suppose that \( f : X \to \mathbb{ R }  \) is a function then 
+        \[
+          \operatorname{ supp } \left( f \right) = \left\{ x \in X : f \left( x \right) \neq 0  \right\} 
+        \] 
+    </div>
+</div>

html/fundamentals/sets/set_difference.html

@@ -161,7 +161,13 @@ <h1>Set Difference</h1>
         DeMorgan's Laws
     </div>
     <div class="content">
-        Suppose that \( M \) is a family of sets, then \( X \backslash \left ( \bigcup_{A \in M} A \right ) = \bigcap_{A \in M} \left ( X \backslash A \right ) \) and \( X \backslash \left ( \bigcap_{A \in M} A \right ) = \bigcup_{A \in M} \left ( X \backslash A \right ) \)
+        Suppose that \( M \) is a family of sets, then 
+        \[
+            X \backslash \left ( \bigcup_{A \in M} A \right ) = \bigcap_{A \in M} \left ( X \backslash A \right ) 
+        \] and 
+        \[ 
+            X \backslash \left ( \bigcap_{A \in M} A \right ) = \bigcup_{A \in M} \left ( X \backslash A \right ) 
+        \]
     </div>
     <div class="proof">
         <p>Note that, \( p \in X \backslash \left ( \bigcup_{A \in M} A \right ) \), if and only if \( p \in X \) and \( x \notin \bigcup_{A \in M} A \), which is equivalent to: for all \( A \in M \) \( p \notin A \), which is the same as \( p \in \left ( X \backslash A \right ) \) for each \( A \in M \), so by definition \( p \in \bigcap_{A \in M} \left ( X \backslash A \right ) \). Since each connective in the previous paragraph was an iff, then this implies the two sets are equal</p>

html/fundamentals/sets/union_and_intersection.html

@@ -3,15 +3,15 @@
         Union
     </div>
     <div class="content">
-        Given two <span class="knowledge-link" data-href="/fundamentals/sets/basics.html#definition-set">sets</span> \( A , B \subseteq X \), then the union of \( A \) and \( B \) is defined as the set \( A \cup B := \left \lbrace p \in X : p \in A \lor p \in B \right \rbrace \)
+        Given two <a class="knowledge-link" href="/fundamentals/sets/basics.html#definition-set">sets</a> \( A , B \subseteq X \), then the union of \( A \) and \( B \) is defined as the set \( A \cup B := \left \lbrace p \in X : p \in A \lor p \in B \right \rbrace \)
     </div>
 </div>
 <div class="definition" id="definition-intersection">
     <div class="title">
         Intersection
     </div>
     <div class="content">
-        Given two <span class="knowledge-link" data-href="/fundamentals/sets/basics.html#definition-set">sets</span> \( A , B \subseteq X \), then the intersection of \( A \) and \( B \) is defined as the set \( A \cap B := \left \lbrace p \in X : p \in A \land p \in B \right \rbrace \)
+        Given two <a class="knowledge-link" href="/fundamentals/sets/basics.html#definition-set">sets</a> \( A , B \subseteq X \), then the intersection of \( A \) and \( B \) is defined as the set \( A \cap B := \left \lbrace p \in X : p \in A \land p \in B \right \rbrace \)
     </div>
 </div>
 <div class="definition" id="definition-a-set-intersects-another">
@@ -43,57 +43,68 @@
         Arbitrary Union Element of Notation
     </div>
     <div class="content">
-        We define \( \bigcup_{A \in M} A := \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup M \)</span>
+        We define \( \bigcup_{A \in M} A := \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup M \)</a>
     </div>
 </div>
 <div class="definition" id="definition-arbitrary-intersection-element-of-notation">
     <div class="title">
         Arbitrary Intersection Element of Notation
     </div>
     <div class="content">
-        We define \( \bigcap_{A \in M} A := \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap M \)</span>
+        We define \( \bigcap_{A \in M} A := \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap M \)</a>
     </div>
 </div>
 <div class="definition" id="definition-arbitrary-union-indexed-notation">
     <div class="title">
         Arbitrary Union Indexed Notation
     </div>
     <div class="content">
-        Suppose that \( I \) is an index set for the collection \( \mathcal{A} = \left \lbrace A_{\alpha} : \alpha \in I \right \rbrace \) where \( A_{\alpha} \) is a set, then \( \bigcup_{\alpha \in I} A_{\alpha} := \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup \mathcal{A} \)</span>. If the index set is known by context, then we may use the shorthand \( \bigcup_{\alpha} A_{\alpha} \)
+        Suppose that \( I \) is an index set for the collection \( \mathcal{A} = \left \lbrace A_{\alpha} : \alpha \in I \right \rbrace \) where \( A_{\alpha} \) is a set, then \( \bigcup_{\alpha \in I} A_{\alpha} := \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup \mathcal{A} \)</a>. If the index set is known by context, then we may use the shorthand \( \bigcup_{\alpha} A_{\alpha} \)
     </div>
 </div>
 <div class="definition" id="definition-arbitrary-intersection-indexed-notation">
     <div class="title">
         Arbitrary Intersection Indexed Notation
     </div>
     <div class="content">
-        Suppose that \( I \) is an index set for the collection \( \mathcal{A} = \left \lbrace A_{\alpha} : \alpha \in I \right \rbrace \) where \( A_{\alpha} \) is a set, then \( \bigcap_{\alpha \in I} A_{\alpha} := \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap \mathcal{A} \)</span>. If the index set is known by context, then we may use the shorthand \( \bigcap_{\alpha} A_{\alpha} \)
+        Suppose that \( I \) is an index set for the collection \( \mathcal{A} = \left \lbrace A_{\alpha} : \alpha \in I \right \rbrace \) where \( A_{\alpha} \) is a set, then \( \bigcap_{\alpha \in I} A_{\alpha} := \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap \mathcal{A} \)</a>. If the index set is known by context, then we may use the shorthand \( \bigcap_{\alpha} A_{\alpha} \)
     </div>
 </div>
 <div class="definition" id="definition-arbitrary-union-counting-notation">
     <div class="title">
         Arbitrary Union Counting Notation
     </div>
     <div class="content">
-        Suppose that \( a , b \in \mathbb{Z} \), with \( a \lt b \), and \( \mathcal{A} := \left \lbrace A_{i} : i \in \mathbb{Z} , a \le i \le b \right \rbrace \), then \( \bigcup_{i = a}^{b} A_{i} = \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup \mathcal{A} \)</span>
+        Suppose that \( a , b \in \mathbb{Z} \), with \( a \lt b \), and \( \mathcal{A} := \left \lbrace A_{i} : i \in \mathbb{Z} , a \le i \le b \right \rbrace \), then \( \bigcup_{i = a}^{b} A_{i} = \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-union">\( \bigcup \mathcal{A} \)</a>
     </div>
 </div>
 <div class="definition" id="definition-arbitrary-intersection-counting-notation">
     <div class="title">
         Arbitrary Intersection Counting Notation
     </div>
     <div class="content">
-        Suppose that \( a , b \in \mathbb{Z} \), with \( a \lt b \), and \( \mathcal{A} := \left \lbrace A_{i} : i \in \mathbb{Z} , a \le i \le b \right \rbrace \), then \( \bigcap_{i = a}^{b} A_{i} = \) <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap \mathcal{A} \)</span>
+        Suppose that \( a , b \in \mathbb{Z} \), with \( a \lt b \), and \( \mathcal{A} := \left \lbrace A_{i} : i \in \mathbb{Z} , a \le i \le b \right \rbrace \), then \( \bigcap_{i = a}^{b} A_{i} = \) <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-arbitrary-intersection">\( \bigcap \mathcal{A} \)</a>
     </div>
 </div>
 <div class="definition" id="definition-disjoint-sets">
     <div class="title">
-        disjoint sets
+        Disjoint Sets
     </div>
     <div class="content">
-        Given two sets \( A , B \) we say that \( A \) and \( B \) are disjoint when <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#definition-intersection">\( A \cap B \)</span> = <span class="knowledge-link" data-href="/fundamentals/sets/basics.html#definition-empty-set">\( \emptyset \)</span>
+        Given two sets \( A , B \) we say that \( A \) and \( B \) are disjoint when <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-intersection">\( A \cap B \)</a> = <a class="knowledge-link" href="/fundamentals/sets/basics.html#definition-empty-set">\( \emptyset \)</a>
     </div>
 </div>
+<div class="definition" id="definition-disjoint-union-notation" >
+    <div class="title">Disjoint Union Notation</div>
+    <div class="content">
+        Given two sets \( A, B \) the notation \( A \sqcup B \) is defined as the set \( A \cup B \) and also that \( A, B \) are <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#definition-disjoint-sets">disjoint</a>.
+    </div>
+</div>
+
+<p>
+    Sometimes the above can feel confusing, because it allows you to write contradictions, ie, if you write \( \left\{ 1 \right\} \sqcup \left\{ 1 \right\}   \) then this is a contradiction, just in the same way as writing \( 1 = 0 \) would also be a contradiction, so it's important to remember that the square union notation is making a claim at the same time so whenever it's utilized you should verify that the two sets are indeed disjoint, and sometimes that will also require proof.
+</p>
+
 <div class="definition" id="definition-pairwise-disjoint-sets">
     <div class="title">
         pairwise disjoint sets
@@ -139,7 +150,7 @@
         Let \( A , B \) be sets such that \( A \subseteq B \), then \( A \cap B = A \)
     </div>
     <div class="proof">
-        <p><span class="knowledge-link" data-href="/fundamentals/sets/basics.html#proposition-set-equality-by-subsets">We start</span> by showing \( A \cap B \subseteq A \), so consider \( x \in A \cap B \), so we know that \( x \in A \) and \( x \in B \), therefore we trivially know that \( x \in A \), as needed</p>
+        <p><a class="knowledge-link" href="/fundamentals/sets/basics.html#proposition-set-equality-by-subsets">We start</a> by showing \( A \cap B \subseteq A \), so consider \( x \in A \cap B \), so we know that \( x \in A \) and \( x \in B \), therefore we trivially know that \( x \in A \), as needed</p>
         <p>Now consider the other direction, we need to show that \( A \subseteq A \cap B \), so consider that \( x \in A \), then we want to show that \( x \in A \) and \( x \in B \), which really just simplifies to showing \( x \in B \), but we know that \( A \subseteq B \), therefore since \( x \in A \), we know that \( x \in B \) finishing the proof</p>
     </div>
 </div>
@@ -260,7 +271,7 @@
         Suppose \( A \) is a set and that for each \( a \in A \), there is a \( B_{a} \) such that \( a \in B_{a} \subseteq A \), then \( A = \bigcup_{a \in A} B_{a} \)
     </div>
     <div class="proof">
-        <p>We can see that \( \mathcal{C} = \left \lbrace B_{a} : a \in A \right \rbrace \) is a collection of subsets of \( A \), so since <span class="knowledge-link" data-href="/fundamentals/sets/union_and_intersection.html#proposition-union-of-subsets-is-still-a-subset">a union of a subsets is still a subset</span> we have \( \bigcup_{a \in A} B_{a} \subseteq A \).</p>
+        <p>We can see that \( \mathcal{C} = \left \lbrace B_{a} : a \in A \right \rbrace \) is a collection of subsets of \( A \), so since <a class="knowledge-link" href="/fundamentals/sets/union_and_intersection.html#proposition-union-of-subsets-is-still-a-subset">a union of a subsets is still a subset</a> we have \( \bigcup_{a \in A} B_{a} \subseteq A \).</p>
         <p>But also given \( p \in A \) we know that \( p \in B_{p} \) so \( p \in \bigcup_{a \in A} B_{a} \) which shows \( A \subseteq \bigcup_{a \in A} B_{a} \), so we can conclude \( A = \bigcup_{a \in A} B_{a} \).</p>
     </div>
 </div>

html/graph_theory/index.html

@@ -0,0 +1,8 @@
+<fieldset>
+    <legend>Graph Theory</legend>
+    <ul>
+        <li><a href="graphs.html">Graphs</a></li>
+        <li><a href="trees.html">Trees</a></li>
+        <li><a href="spectral.html">Spectral</a></li>
+    </ul>
+</fieldset>