vault backup: 2026-03-07 09:46:32

.obsidian/graph.json

@@ -18,5 +18,5 @@
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 }

.obsidian/workspace.json

@@ -20,8 +20,23 @@
               "icon": "lucide-file",
               "title": "Set Theory"
             }
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+                "file": "DAS/Relations.md",
+                "mode": "source",
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+              },
+              "icon": "lucide-file",
+              "title": "Relations"
             }
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@@ -168,12 +183,12 @@
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+    "DAS/Relations.md",
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-    "DAS/Set Theory.md",
     "mathe/notation.md",
     "Studium.md",
     "ET/Netzwerke.md",

DAS/Relations.md

@@ -0,0 +1,40 @@
+## Types of Relations
+
+| Relation         | Explanation                                                                                                           | Example                   |
+| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
+| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$   |
+| *reflexive*      | every element is related to itself with the given relation                                                            | $a <= a, 5 = 5$           |
+| *anti-reflexive* | every element is *NOT* related to itself in the given relation                                                        | $a < a$                   |
+| *symmetric*      | the given relation work both ways                                                                                     | $a = b => b = a$          |
+| *antisymmetric*  | the given relation only works both ways if $a$ and $b$ are the same                                                   | $a <= b, b <= a => a = b$ |
+
+## Equivalence Relations
+A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
+
+### Example: 
+**Question:** How many equivalence classes are there for the given equivalence relation?
+$$
+& ~ "on" {0, 1, 2, 3}^(2) \
+& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
+$$
+> [!INFO]
+> Meaning: 
+> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
+
+Solving: 
+- Smallest possible sum: $(0 + 0) = 0$
+- Biggest possible sum: $(3 + 3) = 6$
+- All possible sums: $0, 1, 2, 3, 4, 5, 6$
+
+Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
+
+> [!NOTE]
+> All equivalence classes:
+> $[0]_(~) = {(0, 0)}$
+> $[1]_(~) = {(0, 1), (1, 0)}$
+> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
+>$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$ 
+>$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
+> $[5]_(~) = {(2, 3), (3, 2)}$
+> $[6]_(~) = {(3, 3)}$
+

DAS/Set Theory.md

@@ -84,12 +84,3 @@ A := {1, 2, 3} \
 cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
 $$
 
-## Relations
-
-| Relation         | Explanation                                                                                                           | Example                   |
-| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
-| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$   |
-| *reflexive*      | every element is related to itself with the given relation                                                            | $a <= a, 5 = 5$           |
-| *anti-reflexive* | every element is *NOT* related to itself in the given relation                                                        | $a < a$                   |
-| *symmetric*      | the given relation work both ways                                                                                     | $a = b => b = a$          |
-| *antisymmetric*  | the given relation only works both ways if $a$ and $b$ are the same                                                   | $a <= b, b <= a => a = b$ |