chore: remove .thrash folder

.obsidian/workspace.json

@@ -13,12 +13,12 @@
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@@ -78,7 +78,7 @@
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.trash/1775572307685 - Lecture 1 Summary.md

@@ -1,18 +0,0 @@
----
-created: 2026-04-07 16:31
-course:
-topic:
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29592872.615316667 - Lecture 1 Recp.md

@@ -1,18 +0,0 @@
----
-created: 2026-04-07 16:32
-course:
-topic:
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29592877 - review.md

@@ -1,18 +0,0 @@
----
-created: 2026-04-07 16:37
-course:
-topic:
-type: lecture
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-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29593223 - 29593223 - Elektrotechnik II.md

@@ -1,19 +0,0 @@
----
-created: 2026-04-07 22:23
-course:
-topic:
-related:
-type: lecture
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-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29593223 - Elektrotechnik II.md

@@ -1,19 +0,0 @@
----
-created: 2026-04-07 22:23
-course:
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-  - university
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-> [!abstract]
->
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----
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-## 📝 Content
-

.trash/29593224 - 29593223 - 29593223 - Elektrotechnik II.md

@@ -1,19 +0,0 @@
----
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-course:
-topic:
-related:
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-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29593226 - ET_II_Folien_gesamt_020426.pdf.md

@@ -1,19 +0,0 @@
----
-created: 2026-04-07 22:26
-course:
-topic:
-related:
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/29593832 - 2026-04-08.md

@@ -1,19 +0,0 @@
----
-created: 2026-04-08 08:32
-course:
-topic:
-related:
-type: lecture
-status: 🔴
-tags:
-  - university
----
-## 📌 Summary
-
-> [!abstract]
->
-
----
-
-## 📝 Content
-

.trash/Untitled 2.md

@@ -1,320 +0,0 @@
-# Arithmetic
-
-## Asymptotic Equivalence Classes (Big-O)
-
-The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
-
-$f asymp g <==> f in O(g) and g in O(f)$
-
-Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
-
-### The "Dominant Term" Rule
-To find which class a function belongs to, simplify it to its core growth rate:
-1.  **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
-2.  **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
-3.  **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
-
-## Euclidian Algorithm
-Purpose is to find the **GCD** (Greatest Common Divisor).
-
-### Core Rule
-Fill out this formula:
-$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
-1. **Divide** the bigger number by the smaller one
-2. **How many times** does it fit -> $"Quotient"$
-3. Find out **whats leftover** -> $"Remainder"$
-4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
-
-Result is the last $"Remainder"$ that is not $0$.
-
-### Bezout Coefficients
-**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
-
-1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
-2.  **Substitute remainders** -> $$
-
----
-
-# Counting
-
-## Inclusion-Exclusion Principle
-Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
-
-### Example:
-How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
-#### How many $x^2$?
-$sqrt(10^6) = 10^3 = 1.000$
-#### How many $x^5$?
-By estimation:
-$$
-&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"
-$$
-#### Final calculation:
-Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
-$==> 1,000 + 15 - 3 = 1,012$
-
-### For two sets
-$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
-Where 
-- $"Avoid 1"$ are all elements *not matching* condition 1
-- $"Avoid 2"$ are all elements *not matching* condition 2
-- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
-
-## Type of Task: Boolean Lattice
-Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
-$$
-& x = (0, 1, 0, 0, 0) \
-& y = (0, 0, 1, 0, 1) \ 
-& z = (0, 1, 1, 0, 0) \
-$$
-> [!INFO]
-> Method used is called **All-Zero Column Method**
-
-We look for columns that have $0$'s in all three vectors: Column $1$ and $4$, that's $2$ columns, so we define $k = 2$.
-To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$: 
-$2^2 = 4$.
-So we have **4** upper bounds in total.
-
-> [!INFO]
-> For the amount of lower bounds we'd check for all-ones columns
-
----
-
-# Functions
-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$
-
----
-
-# Logic
-
-## Operators 
-| Operation         | Explanation                          | Notation  |
-| ----------------- | ------------------------------------ | --------- |
-| **and**<br>       | Both $p$ and $q$ must be true        | $p and q$ |
-| **or** | Either $p$ or $q$ (or both) are true | $p or q$  |
-| **not** | Negates the statement                | $not p$   |
-| **Implication** | If $p$ then $q$                      | $=>$      |
-| **Biconditional** | $p$ if and _only_ if $q$             | $<=>$     |
-| **xor** | Either $p$ or $q$ but not both       | $xor$     |
-
-### Implied Operators
-| Operation | Explanion                               | Notation       |
-| --------- | --------------------------------------- | -------------- |
-| **nand** | $p$ and $q$ are not both true           | $not(p and q)$ |
-| **nor** | neither of $p$ and $q$ are true         | $not(p or q)$  |
-| **xnor** | $p$ and $q$ are both false or both true | $not xor$      |
-
----
-
-# Relations
-
-## Types of Relations
-
-| Relation         | Explanation                                                                                                           | Example                   |
-| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
-| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$   |
-| *reflexive* | every element is related to itself with the given relation                                                            | $a <= a, 5 = 5$           |
-| *anti-reflexive* | every element is *NOT* related to itself in the given relation                                                        | $a < a$                   |
-| *symmetric* | the given relation work both ways                                                                                     | $a = b => b = a$          |
-| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same                                                   | $a <= b, b <= a => a = b$ |
-
-
-## Equivalence Relations
-A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
-
-### Example: 
-**Question:** How many equivalence classes are there for the given equivalence relation?
-$$
-& ~ "on" {0, 1, 2, 3}^(2) \
-& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
-$$
-> [!INFO]
-> Meaning: 
-> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
-
-Solving: 
-- Smallest possible sum: $(0 + 0) = 0$
-- Biggest possible sum: $(3 + 3) = 6$
-- All possible sums: $0, 1, 2, 3, 4, 5, 6$
-
-Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
-
-> [!NOTE]
-> All equivalence classes:
-> $[0]_(~) = {(0, 0)}$
-> $[1]_(~) = {(0, 1), (1, 0)}$
-> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
-> $[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$ 
-> $[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
-> $[5]_(~) = {(2, 3), (3, 2)}$
-> $[6]_(~) = {(3, 3)}$
-
-## Binary Relation
-A binary relation is a relation $R$ between _exactly two_ elements $a in R$ and $b in R$. An example for a binary relation is $a <= b$
-
-## Converse Relation
-$C^top$ or $C^(-1)$ is the relation that occurs if the elements of a _binary relation_ are switched.
-
-## Composition of Relations - Example
-$$
-"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)}
-$$
-### 2. Perform Composition:
-Look at each pair in $R$, check if $Q^top$ has a pair starting with se second element in that pair:
-
-$$
-(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)}$$
-
-## Orders
-An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.
-
-A _binary relation_ is called an order if it is...
-- [?] a *reflexive relation*
-- [?] a *antisymmetric relation*
-- [?] a *transitive relation*
-
----
-
-# Set Theory
-A set is a collection of _unordered_ elements.
-A set cannot contain duplicates.
-
-## Notation
-### Set Notation
-Declaration of a set $A$ with elements $a$, $b$, $c$: 
-$$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}}
-$$

.trash/mathe/notation.md

@@ -1,5 +0,0 @@
-# Notation
-
-## 1. Sets and Logic
-$|A|$: The *cardinality* (size) of finite set $A$.
-$script(p)$

.trash/merge.md

@@ -1,284 +0,0 @@
-## Asymptotic Equivalence Classes (Big-O)
-
-The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
-
-$f asymp g <==> f in O(g) and g in O(f)$
-
-Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
-
-###  The "Dominant Term" Rule
-To find which class a function belongs to, simplify it to its core growth rate:
-1.  **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
-2.  **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
-3.  **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
-
-## Euclidian Algorithm
-Purpose is to find the **GCD** (Greatest Common Divisor).
-
-### Core Rule
-Fill out this formula:
-$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
-1. **Divide** the bigger number by the smaller one
-2. **How many times** does it fit -> $"Quotient"$
-3. Find out **whats leftover** -> $"Remainder"$
-4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
-
-Result is the last $"Remainder"$ that is not $0$.
-
-### Bezout Coefficients
-**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
-
-1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
-2.  **Substitute remainders** -> ## 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: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
-$==> 1,000 + 15 - 3 = 1,012$
-
-### For two sets
-$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
-Where 
-- $"Avoid 1"$ are all elements *not matching* condition 1
-- $"Avoid 2"$ are all elements *not matching* condition 2
-- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
-
-## Type of Task: Boolean Lattice
-Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
-$$
-& x = (0, 1, 0, 0, 0) \
-& y = (0, 0, 1, 0, 1) \ 
-& z = (0, 1, 1, 0, 0) \
-$$
-> [!INFO]
-> Method used is called **All-Zero Column Method**
-
-We look for columns that have $0$'s in all three vectors: Column $1$ and $4$, that's $2$ columns, so we define $k = 2$.
-To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$: 
-$2^2 = 4$.
-So we have **4** upper bounds in total.
-
-> [!INFO]
-> For the amount of lower bounds we'd check for all-ones columns
-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_)
-- $forall y in Y, exists excl x in X : f(x) = y$## Operators 
-| Operation         | Explanation                          | Notation  |
-| ----------------- | ------------------------------------ | --------- |
-| **and**<br>       | Both $p$ and $q$ must be true        | $p and q$ |
-| **or**            | Either $p$ or $q$ (or both) are true | $p or q$  |
-| **not**           | Negates the statement                | $not p$   |
-| **Implication**   | If $p$ then $q$                      | $=>$      |
-| **Biconditional** | $p$ if and _only_ if $q$             | $<=>$     |
-| **xor**           | Either $p$ or $q$ but not both       | $xor$     |
-### Implied Operators
-| Operation | Explanion                               | Notation       |
-| --------- | --------------------------------------- | -------------- |
-| **nand**  | $p$ and $q$ are not both true           | $not(p and q)$ |
-| **nor**   | neither of $p$ and $q$ are true         | $not(p or q)$  |
-| **xnor**  | $p$ and $q$ are both false or both true | $not xor$      |
-## Types of Relations
-
-| Relation         | Explanation                                                                                                           | Example                   |
-| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
-| *transitive*<br> | "chain reaction", a information about $a$ in relation to $c$ can be inferred from the relations $a -> b$ and $b -> c$ | $a < b, b < c => a < c$   |
-| *reflexive*      | every element is related to itself with the given relation                                                            | $a <= a, 5 = 5$           |
-| *anti-reflexive* | every element is *NOT* related to itself in the given relation                                                        | $a < a$                   |
-| *symmetric*      | the given relation work both ways                                                                                     | $a = b => b = a$          |
-| *antisymmetric*  | the given relation only works both ways if $a$ and $b$ are the same                                                   | $a <= b, b <= a => a = b$ |
-
-
-## Equivalence Relations
-A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
-
-### Example: 
-**Question:** How many equivalence classes are there for the given equivalence relation?
-$$
-& ~ "on" {0, 1, 2, 3}^(2) \
-& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
-$$
-> [!INFO]
-> Meaning: 
-> The pairs $(x_1, y_1)$ and $(x_2, y_2)$ are equivalent to each other when the components of the pair added up have the same result.
-
-Solving: 
-- Smallest possible sum: $(0 + 0) = 0$
-- Biggest possible sum: $(3 + 3) = 6$
-- All possible sums: $0, 1, 2, 3, 4, 5, 6$
-
-Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
-
-> [!NOTE]
-> All equivalence classes:
-> $[0]_(~) = {(0, 0)}$
-> $[1]_(~) = {(0, 1), (1, 0)}$
-> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
->$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$ 
->$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
-> $[5]_(~) = {(2, 3), (3, 2)}$
-> $[6]_(~) = {(3, 3)}$
-
-## Binary Relation
-A binary relation is a relation $R$ between _exactly two_ elements $a in R$ and $b in R$. An example for a binary relation is $a <= b$
-
-## Converse Relation
-$C^top$ or $C^(-1)$ is the relation that occurs if the elements of a _binary relation_ are switched.
-
-## Composition of Relations - Example
-$$
-"Compute" Q^top compose R "with:"\
-Q = {(2, 2), (3, 3), (2, 1)} \
-R = {(1, 2), (3, 3), (3, 1)} \
-$$
-### 1. Apply converse to $Q$:
-$$
-Q^top = {(2, 2), (3, 3), (1, 2)}
-$$
-### 2. 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.
-
-A _binary relation_ is called an order if it is...
-- [?] a *reflexive relation*
-- [?] a *antisymmetric relation*
-- [?] a *transitive relation*
-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_
-$$
-A := {1, 2, 3} \
-cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
-$$
-

00 Inbox/29593852 - Strings.md

@@ -18,37 +18,91 @@ tags:
 ## 📝 Content
 
 ### Alphabets
-Alphabets are formal, non-empty, sets of symbols (usually lowercase letters). They are denoted by $Sigma$.
+Alphabets are formal, non-empty, finite, sets of characters (or _letters_ or _symbols_). They are denoted by $Sigma$.
 
 $Sigma = {a, b}$
 > Alphabet $Sigma$ contains the characters $a$ and $b$.
 
+$Sigma = {a, ..., z, A, ..., Z, 0, ..., 9}$
+> usual alphabet for writing text
+
 ### Strings
-A string is a set of letters. If there is an alphabet $Sigma = {a,b}$ then `abba` is a string made from that alphabet.
+A word (or _string_) is a finite sequence $w = a_1 a_2 ... a_n$ if characters from $Sigma$.
+
+> [!CONVENTION]
+> We will use small letters to describe strings that are part of a language.
+
+> [!EXAMPLE]
+> $"aa", "ab", "bba"$ and $"baab"$ are strings over $Sigma = {a, b}.
+
+#### Length of a string
+The _length_ $abs(x)$ of a string $x = a_1 ... a_n$ is its number $abs(x) = n$ of characters.
+#### Empty String
 The empty string is denoted by $epsilon$, this is the neutral element.
+-> $abs(epsilon) = 0$
 ### Concatenation
-String can be concatenated, where one string is appended to another.
-$"apple" dot "pie" = "applepie"$
+String can be concatenated, where one string is appended to another. 
+For strings $x = a_1 ... a_n$ and $y  = b_1 ... b_m$ over alphabets $Sigma_x$ and $Sigma_y$, their _concatenation_ over the alphabet $Sigma = Sigma_x union Sigma_y$ is the string
+$$x circle.small y = x y = a_1 a_2 ... a_n b_1 b_2 ... b_m$$
+> This string is of the length $abs(x y) = n + m$
 
-$&x = "apple" \ &y = "pie" \ &x dot y = "applepie"$
+> [!EXAMPLE]
+> $x = "apple"$
+> $y = "pie"$ 
+> $x circle.small y = "applepie"$
+
+Order of operations / Brackets do _not matter_. (Concatenation is associative but **not** commutative $x y eq.not y x$)
+> $(x circle.small y) circle.small z = x circle.small (y circle.small z)$
 
-Order of operations / Brackets do _not matter_.
 Any string concatenated with the empty string $epsilon$ will result in itself.
-
+> $x circle.small epsilon = x = epsilon circle.small x$ 
 ### Exponentiation
 
+The $n^"th"$ power $x^n$ of a string $x$ is the $(n-1)$-fold concatenation of $x$ with itself.
+> $x^0 := epsilon$
+> $x^n := x^(n-1) circle.small x$ for $n in NN$
+
+> [!Example]
+> $x^4 = x x x x$
+> $(a b)^3 = a b a b a b$
 
 ### Reversing / Mirroring
 For a string $x = a_1 a_2 ... a_(n-1) a_n$ of length $n$, it's _mirrored string_ is given by 
 $$ x^("Rev") = a_n a_(n-1)...a_2 a_1$$
+### Substrings
+A string $x$ is a _substring_ of a string $y$ if $y = u x v$, where $u$ and $v$ can be arbitrary strings.
+- If $u = epsilon$ then $x$ is a _prefix_ of $y$.
+- If $v = epsilon$ then $x$ is a suffix of $y$.
+
+For strings $x$ and $y$ the quantity $abs(y)_x$ is the number of times that $x$ is a substring of $y$.
 
 ### Kleene Star
-Denoted by $Sigma^*$. The Kleene Star (or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet.
+Denoted by $Sigma^*$. The Kleene Star (or _Kleene operator_ or _Kleene Closure_) gives an infinite amount of strings made up of the characters of the alphabet $Sigma ^ *$. 
+$Sigma^*$ is the set of all string that can be generated by arbitrary concatenation of its characters.
+> $Sigma^* := union.big_(n>=0) A_n$
+> where $A_n$ is the set of all string combinations of length $n$
 
-$$
-Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}
-$$
-> Example of the Kleene Star of the alphabet ${a, b}$
+#### Remarks
+- The same character can be used multiple times.
+- The empty string $epsilon$ is also part f $Sigma^*$.
+
+> [!Example]
+> $Sigma^* {a, b} = {epsilon, a, b, "aa", "ab", "ba", "bb", "aaa", "aab", ...}$
+
+> [!FACT]
+> - The set $Sigma^*$ is infinite, since we defined $Sigma$ to be non-empty.
+> - It is _countable_ and has the same cardinality as the set $NN$ of natural numbers
+
+#### Kleene Plus
+The _Kleene Plus_ of an alphabet $Sigma$ is given by $Sigma^+ = Sigma^* backslash {epsilon}$ 
+
+#### Lemma group structure
+The structure _Lemma_ is induced by the Kleene star - it is a monoid, that is a semigroup with a neutral element.
+
+> [!PROOF]
+> - Associativity has been shown
+> - Existence  of a neutral element has been shown.
+> - Closure under $circle.small$: Let $x in Sigma^*$ and $y in Sigma^*$ be two string over the alphabet $Sigma$. Then $x circle.small y = x y in Sigma^*$
 ### Formal Languages
 A formal _language_ of the alphabet $Sigma$ is a subset $L$ of $Sigma^*$
 

00 Inbox/29593929 - Alphabets.md

@@ -0,0 +1,18 @@
+---
+created: 2026-04-08 10:09
+course: "[[29593850 - Automationtheory]]"
+topic: "#alphabets"
+related:
+type: lecture
+status: 🔴
+tags:
+  - university
+---
+	## 📌 Summary
+
+> [!abstract]
+> 
+
+---
+
+## 📝 Content