vault backup: 2026-03-09 18:11:18

.obsidian/workspace.json

@@ -34,9 +34,23 @@
               "icon": "lucide-file",
               "title": "Functions"
             }
+          },
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+            "type": "leaf",
+            "state": {
+              "type": "markdown",
+              "state": {
+                "file": "Counting.md",
+                "mode": "source",
+                "source": false
+              },
+              "icon": "lucide-file",
+              "title": "Counting"
+            }
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@@ -194,12 +208,13 @@
       "obsidian-git:Open Git source control": false
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+  "active": "d840acc1de60406b",
   "lastOpenFiles": [
+    "DAS/Functions.md",
+    "Counting.md",
     "DAS/Relations.md",
     "DAS/Logic.md",
     "DAS/Set Theory.md",
-    "DAS/Functions.md",
     "mathe/notation.md",
     "Studium.md",
     "ET/Netzwerke.md",

Counting.md

@@ -0,0 +1,26 @@
+## Inclusion-Exclusion Principle
+ Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
+
+### Example:
+How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
+#### How many $x^2$?
+$sqrt(10^6) = 10^3 = 1.000$
+#### How many $x^5$?
+By estimation:
+$$
+&15^5 = 759,375 && "--- in the range / below" 10^6 \
+&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
+&=> 15 "numbers in the form "x^5"exist" &&
+$$
+> [!warning]
+> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
+
+#### How many $x^2$ and $x^5$ / $x^10$?
+$$
+& 3^10 = 59,049 \
+& 4^10 = 1,048,576 \
+& => 3 "numbers that are both" x^2 "and" x^5 "exist"
+$$
+#### Final calculation:
+Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
+$==> 1,000 + 15 - 3 = 1,012$