vault backup: 2026-04-07 13:34:22

10 Courses/01 - WiSe 2025_26/DAS/Functions.md

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+A function takes in an element from its **domain**, transforms it in someway and outputs that transformed element, which is part of the **co-domain**.
+Every element of the **domain** must map to a value in the **co-domain**, all values of the **co-domain** that are mapped to form the functions **image**.
+A function must be **deterministic** - one input can only map to a single output.
+
+## Notation
+General function notation: $f: X -> Y$
+
+> [!INFO]
+> $f$: name of the function
+> $X$: Domain
+> $Y$: Co-domain
+> $f(x)$: Image of $f$
+> $X$, $f(x)$ and $Y$ are [[Set Theory | Set]]
+
+For any $x in X$ the output $f(x)$ is an element of $Y$.
+
+## Mapping Properties
+### 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 $x in X$
+- $forall y in Y, exists x in X : f(x) = y$
+
+### Bijectivity
+A function is _bijective_ if every element $y in Y$ has _exactly_ one matching $x in X$ (it is _injective_ and _surjective_)
+- $forall y in Y, exists excl x in X : f(x) = y$