diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 3435719..34e7cc6 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -27,12 +27,12 @@ "state": { "type": "markdown", "state": { - "file": "DAS/Relations.md", + "file": "DAS/Functions.md", "mode": "source", "source": false }, "icon": "lucide-file", - "title": "Relations" + "title": "Functions" } } ], @@ -196,9 +196,9 @@ }, "active": "16b88a8bc6c6b6e0", "lastOpenFiles": [ + "DAS/Relations.md", "DAS/Logic.md", "DAS/Set Theory.md", - "DAS/Relations.md", "DAS/Functions.md", "mathe/notation.md", "Studium.md", diff --git a/DAS/Functions.md b/DAS/Functions.md index 62c676e..32b29ed 100644 --- a/DAS/Functions.md +++ b/DAS/Functions.md @@ -18,7 +18,6 @@ For any $x in X$ the output $f(x)$ is an element of $Y$. ### Injectivity A function is _injective_ if every element in $y in f(x)$ has _at most_ one matching $x in X$. - $forall y in Y,exists excl x in X : f(x) = y$ - ### Surjectivity A function is _surjective_ if every element $y in Y$ has _at minimum_ one matching $x in X$ - $forall y in Y, exists x in X : f(x) = y$ diff --git a/DAS/Relations.md b/DAS/Relations.md index 1e18d33..62e7f6d 100644 --- a/DAS/Relations.md +++ b/DAS/Relations.md @@ -50,12 +50,21 @@ $$ "Compute" Q^top compose R "with:"\ Q = {(2, 2), (3, 3), (2, 1)} \ R = {(1, 2), (3, 3), (3, 1)} \ -\ -"1. Apply converse to Q:"\ -Q^top = {(2, 2), (3, 3), (1, 2)} - -"2. Check " $$ +### 1. Apply converse to $Q$: +$$ +Q^top = {(2, 2), (3, 3), (1, 2)} +$$ +### 2. Perform Composition: +Look at each pair in $R$, check if $Q^top$ has a pair starting with se second element in that pair: + +$$ +(1, 2) -> (2, 2) => (1, 2) \ +(3, 3) -> (3, 3) => (3, 3) \ +(3, 1) -> (1, 2) => (3, 2) +$$ +### 3. Result: +$$ Q^top compose R = {(1, 2), (3, 2), (3, 3)} $$ ## Orders An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.