diff --git a/.obsidian/app.json b/.obsidian/app.json
index 769a3c3..b3ea143 100644
--- a/.obsidian/app.json
+++ b/.obsidian/app.json
@@ -1,5 +1,11 @@
{
"trashOption": "local",
"vimMode": true,
- "alwaysUpdateLinks": true
+ "alwaysUpdateLinks": true,
+ "pdfExportSettings": {
+ "pageSize": "A4",
+ "landscape": false,
+ "margin": "2",
+ "downscalePercent": 100
+ }
}
\ No newline at end of file
diff --git a/.obsidian/appearance.json b/.obsidian/appearance.json
index 9be5bf4..e37e8c4 100644
--- a/.obsidian/appearance.json
+++ b/.obsidian/appearance.json
@@ -1,5 +1,8 @@
{
"theme": "obsidian",
"interfaceFontFamily": "Noto Sans",
- "cssTheme": "Tokyo Night"
+ "cssTheme": "Tokyo Night",
+ "enabledCssSnippets": [
+ "export"
+ ]
}
\ No newline at end of file
diff --git a/.obsidian/snippets/export.css b/.obsidian/snippets/export.css
new file mode 100644
index 0000000..dab8706
--- /dev/null
+++ b/.obsidian/snippets/export.css
@@ -0,0 +1,70 @@
+/* ULTRA-COMPACT EXAM CHEAT SHEET MODE */
+@media print {
+
+ /* 1. Obliterate Page Margins */
+ @page {
+ margin: 0.5cm;
+ size: A4 portrait;
+ }
+
+ /* 2. Tiny but readable base font */
+ body {
+ font-size: 8pt !important;
+ line-height: 1.1 !important;
+ }
+
+ /* 3. Force 3-Column Layout */
+ .markdown-rendered {
+ column-count: 3;
+ column-gap: 0.4cm;
+ column-rule: 1px dashed #ccc;
+ }
+
+ /* 4. Crush Headings and Whitespace */
+ h1,
+ h2,
+ h3,
+ h4,
+ h5 {
+ font-size: 9pt !important;
+ margin: 4px 0 2px 0 !important;
+ padding: 0 !important;
+ line-height: 1.1 !important;
+ }
+
+ p,
+ ul,
+ ol,
+ blockquote {
+ margin: 0 0 3px 0 !important;
+ padding-left: 12px !important;
+ }
+
+ /* 5. Shrink Tables & Callouts */
+ table {
+ font-size: 7pt !important;
+ margin: 2px 0 !important;
+ }
+
+ th,
+ td {
+ padding: 2px 4px !important;
+ }
+
+ .callout {
+ padding: 4px !important;
+ margin: 2px 0 !important;
+ }
+
+ .callout-title {
+ font-size: 8pt !important;
+ }
+
+ /* 6. Prevent awkward page breaks in the middle of math/tables */
+ table,
+ img,
+ .math-block,
+ .callout {
+ break-inside: avoid;
+ }
+}
diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 2d04ed0..452dae7 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -69,12 +69,12 @@
"state": {
"type": "markdown",
"state": {
- "file": "Formulas.md",
+ "file": "DAS/merge.md",
"mode": "source",
"source": false
},
"icon": "lucide-file",
- "title": "Formulas"
+ "title": "merge"
}
}
],
@@ -238,13 +238,15 @@
},
"active": "42deeb92efbbb47c",
"lastOpenFiles": [
+ "DAS/export.css",
"DAS/Arithmetic.md",
+ "DAS/merge.md",
+ "DAS/Set Theory.md",
"Formulas.md",
"DAS/Functions.md",
"DAS/Counting.md",
"DAS/Relations.md",
"DAS/Logic.md",
- "DAS/Set Theory.md",
"mathe/notation.md",
"Studium.md",
"ET/Netzwerke.md",
diff --git a/DAS/Arithmetic.md b/DAS/Arithmetic.md
index 2b4587e..0ed6c19 100644
--- a/DAS/Arithmetic.md
+++ b/DAS/Arithmetic.md
@@ -29,6 +29,4 @@ Result is the last $"Remainder"$ that is not $0$.
**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** -> $$
-
-
+2. **Substitute remainders** ->
\ No newline at end of file
diff --git a/DAS/merge.md b/DAS/merge.md
new file mode 100644
index 0000000..79d1b1c
--- /dev/null
+++ b/DAS/merge.md
@@ -0,0 +1,284 @@
+## Asymptotic Equivalence Classes (Big-O)
+
+The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
+
+$f asymp g <==> f in O(g) and g in O(f)$
+
+Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
+
+### The "Dominant Term" Rule
+To find which class a function belongs to, simplify it to its core growth rate:
+1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
+2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
+3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
+
+## Euclidian Algorithm
+Purpose is to find the **GCD** (Greatest Common Divisor).
+
+### Core Rule
+Fill out this formula:
+$$"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**
| 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*
| "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}
+$$
+### Member of
+Denote whether $x$ is an element of the set $A$
+Notation: $x in A$
+Negation: $x in.not A$
+
+### Subsets
+| Type | Explanation | Notation |
+| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
+| **Subset** | Every element of $A$ is in $B$ | $A subset B$ |
+| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ |
+| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$
|
+| **Superset**
| $A$ contains everything that is in $B$ | $A supset B$ |
+| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ |
+
+## Operations
+### Union
+Notation: $A union B$
+Definition: all elements from both sets _without adding duplicates_
+$$ 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}}
+$$
+
diff --git a/Formulas.md b/Formulas.md
index e69de29..6289ca2 100644
--- a/Formulas.md
+++ b/Formulas.md
@@ -0,0 +1,288 @@
+## Asymptotic Equivalence Classes (Big-O)
+
+The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
+
+$f asymp g <==> f in O(g) and g in O(f)$
+
+Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
+
+### The "Dominant Term" Rule
+To find which class a function belongs to, simplify it to its core growth rate:
+1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
+2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
+3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
+
+## Euclidian Algorithm
+Purpose is to find the **GCD** (Greatest Common Divisor).
+
+### Core Rule
+Fill out this formula:
+$$"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**
| 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*
| "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 $B$ | $A subset.sq B$
|
+| **Superset**
| $A$ 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}}