From e090eadcfae67f509f8e04abe1cae798203129a7 Mon Sep 17 00:00:00 2001 From: Jan Meyer Date: Mon, 9 Mar 2026 19:43:13 +0100 Subject: [PATCH] vault backup: 2026-03-09 19:43:13 --- .obsidian/app.json | 8 +- .obsidian/appearance.json | 5 +- .obsidian/snippets/export.css | 70 +++++++++ .obsidian/workspace.json | 8 +- DAS/Arithmetic.md | 4 +- DAS/merge.md | 284 +++++++++++++++++++++++++++++++++ Formulas.md | 288 ++++++++++++++++++++++++++++++++++ 7 files changed, 659 insertions(+), 8 deletions(-) create mode 100644 .obsidian/snippets/export.css create mode 100644 DAS/merge.md 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}}