diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 34e7cc6..dda036f 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -34,9 +34,23 @@ "icon": "lucide-file", "title": "Functions" } + }, + { + "id": "d840acc1de60406b", + "type": "leaf", + "state": { + "type": "markdown", + "state": { + "file": "Counting.md", + "mode": "source", + "source": false + }, + "icon": "lucide-file", + "title": "Counting" + } } ], - "currentTab": 1 + "currentTab": 2 } ], "direction": "vertical" @@ -194,12 +208,13 @@ "obsidian-git:Open Git source control": false } }, - "active": "16b88a8bc6c6b6e0", + "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", diff --git a/Counting.md b/Counting.md new file mode 100644 index 0000000..9c3573e --- /dev/null +++ b/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: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$ +$==> 1,000 + 15 - 3 = 1,012$