vault backup: 2026-03-09 23:52:22

.obsidian/workspace.json

@@ -250,13 +250,13 @@
       "obsidian-git:Open Git source control": false
     }
   },
-  "active": "7a0e7b37bd89861d",
+  "active": "7d1b3d98021093a7",
   "lastOpenFiles": [
-    "DAS/Arithmetic.md",
+    "DAS/Set Theory.md",
     "DAS/Groups.md",
+    "DAS/Arithmetic.md",
     "DAS/merge.md",
     "DAS/export.css",
-    "DAS/Set Theory.md",
     "Formulas.md",
     "DAS/Functions.md",
     "DAS/Counting.md",

DAS/Groups.md

@@ -1,3 +1,7 @@
+Groups are Closures that have the following properties:
+
+1. **closure**: Every element which is a result of the binary operation of the group is part of the group
+2. **associativity**: changing the order of operations doesn't change the result
+3. **identity**: there is a neutral element ($0$ in addition, $1$ in multiplication)
+4. **inverses**: every element has a element when combined results in the identity element ($+5 + (-5) = 0$)
    
-## Internal Composition law
-$$

DAS/Set Theory.md

@@ -84,3 +84,8 @@ A := {1, 2, 3} \
 cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
 $$
 
+## Closures
+A **closed set** under the binary operation $*$ denoted by $forall a,b in S, a * b in S$
+"All $a * b$ from set $S$ are included in set $S$"
+
+A set can be closed under any binary operation ($+, -, *, \/$), depending on the set itself if it actually is closed or not. $NN$ for example is not closed under subtraction.