chore: inital commit

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 $Y$ has _at most_ one matching $x in X$.
+- $forall y in Y,exists^(<=1) x in X : f(x) = y$
+
+### 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_)

DAS/Set Theory.md

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+> A set is a collection of _unordered_ elements.
+> A set cannot contain duplicates.
+## Notation
+### Set Notation
+Declaration of a set $A$ with elements $a$, $b$, $c$: 
+$$ A := {a, b, c} $$
+### Cardinality
+Amount of Elements in a set $A$
+Notation: $|A|$
+$$
+A := {1, 2, 3, 4} \
+|A| = 4
+$$
+### Well-Known Sets
+- Empty Set: $emptyset = {}$
+- Natural Numbers: $N = {1, 2, 3, ...}$
+- Integers: $ZZ = {-2, -1, 0, 1, 2}$
+- Rational Numbers: $QQ = {1/2, 22/7 }$
+- Real Numbers: $RR = {1, pi, sqrt(2)}$
+- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$
+
+### Set-Builder Notation
+Common form of notation to create sets without explicitly specifying elements.
+$$
+A := {x in N | 0 <= x <= 5} \
+A = {1, 2, 3, 4, 5}
+$$
+### Member of
+Denote whether $x$ is an element of the set $A$
+Notation: $x in A$
+Negation: $x in.not A$
+
+### Subsets
+| Type                     | Explanation                                                            | Notation            |
+| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
+| **Subset**               | Every element of $A$ is in $B$                                         | $A subset B$        |
+| **Subset or equal to**   | Every element of $A$ is in $B$, or they are the exactly same set       | $A subset.eq B$     |
+| **Proper subset**        | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
+| **Superset**<br>         | $A$ contains everything that is in $B$                                 | $A supset B$        |
+| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical          | $A supset.eq B$     |
+
+## Operations
+### Union
+Notation: $A union B$
+Definition: all elements from both sets _without adding duplicates_
+$$ A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5} $$
+### Intersection
+Notation:$A inter B$
+Definition: all elements _contained in both sets_
+$$
+A := {1, 2, 3} \
+B := {2, 3, 4} \
+A inter B = {2, 3}
+$$
+### Difference
+Notation: $A backslash B$ 
+Definition: all elements _in $A$ that are not in $B$_
+$$
+A := {1, 2, 3} \
+B := {3, 4, 5} \
+A backslash B = {1, 2}
+$$ 
+### Symmetric Difference
+Notation: $A Delta B$ 
+Definition: all elements _only in $A$ or only in $B$_
+$$
+A := {1, 2, 3} \
+B := {2, 3, 4} \
+A Delta B = {1, 4}
+$$
+### Cartesian Product
+Notation: $A times B$
+Definition: all pairs of all elements in $A$ and $B$
+$$
+A := {1, 2} \
+B := {3, 4, 5} \
+A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
+$$
+### Powerset
+Notation: $cal(P)(A)$
+Definition: all possible _Subsets of A_
+$$
+A := {1, 2, 3} \
+cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
+$$