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Lecture 1 Summary.md +++ /dev/null @@ -1,18 +0,0 @@ ---- -created: 2026-04-07 16:31 -course: -topic: -type: lecture -status: 🔴 -tags: - - university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/29592872.615316667 - Lecture 1 Recp.md b/.trash/29592872.615316667 - Lecture 1 Recp.md deleted file mode 100644 index c4b1833..0000000 --- a/.trash/29592872.615316667 - Lecture 1 Recp.md +++ /dev/null @@ -1,18 +0,0 @@ ---- -created: 2026-04-07 16:32 -course: -topic: -type: lecture -status: 🔴 -tags: - - university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/29592877 - review.md b/.trash/29592877 - review.md deleted file mode 100644 index dac4b23..0000000 --- a/.trash/29592877 - review.md +++ /dev/null @@ -1,18 +0,0 @@ ---- -created: 2026-04-07 16:37 -course: -topic: -type: lecture -status: 🔴 -tags: - - university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/29593223 - 29593223 - Elektrotechnik II.md b/.trash/29593223 - 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- university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/29593226 - ET_II_Folien_gesamt_020426.pdf.md b/.trash/29593226 - ET_II_Folien_gesamt_020426.pdf.md deleted file mode 100644 index 98e626d..0000000 --- a/.trash/29593226 - ET_II_Folien_gesamt_020426.pdf.md +++ /dev/null @@ -1,19 +0,0 @@ ---- -created: 2026-04-07 22:26 -course: -topic: -related: -type: lecture -status: 🔴 -tags: - - university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/29593832 - 2026-04-08.md b/.trash/29593832 - 2026-04-08.md deleted file mode 100644 index c2c7960..0000000 --- a/.trash/29593832 - 2026-04-08.md +++ /dev/null @@ -1,19 +0,0 @@ ---- -created: 2026-04-08 08:32 -course: -topic: -related: -type: lecture -status: 🔴 -tags: - - university ---- -## 📌 Summary - -> [!abstract] -> - ---- - -## 📝 Content - diff --git a/.trash/Online test.md b/.trash/Online test.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Test File.md b/.trash/Test File.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Untitled 2.md b/.trash/Untitled 2.md deleted file mode 100644 index 38da010..0000000 --- a/.trash/Untitled 2.md +++ /dev/null @@ -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**
| 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*
| "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} -$$ - -### Member of -Denote whether $x$ is an element of the set $A$ -Notation: $x in A$ -Negation: $x in.not A$ - -### Subsets -| Type | Explanation | Notation | -| ------------------------ | ---------------------------------------------------------------------- | ------------------- | -| **Subset** | Every element of $A$ is in $B$ | $A subset B$ | -| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ | -| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$
| -| **Superset**
| $A$ contains everything that is in $B$ | $A supset B$ | -| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ | - -## Operations -### Union -Notation: $A union B$ -Definition: all elements from both sets _without adding duplicates_ -$$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}} -$$ \ No newline at end of file diff --git a/.trash/Untitled 3.md b/.trash/Untitled 3.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Untitled 4.md b/.trash/Untitled 4.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Untitled 5.md b/.trash/Untitled 5.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Untitled 6.md b/.trash/Untitled 6.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/Untitled.md b/.trash/Untitled.md deleted file mode 100644 index e69de29..0000000 diff --git a/.trash/mathe/notation.md b/.trash/mathe/notation.md deleted file mode 100644 index de64d79..0000000 --- a/.trash/mathe/notation.md +++ /dev/null @@ -1,5 +0,0 @@ -# Notation - -## 1. Sets and Logic -$|A|$: The *cardinality* (size) of finite set $A$. -$script(p)$ diff --git a/.trash/merge.md b/.trash/merge.md deleted file mode 100644 index 79d1b1c..0000000 --- a/.trash/merge.md +++ /dev/null @@ -1,284 +0,0 @@ -## Asymptotic Equivalence Classes (Big-O) - -The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$): - -$f asymp g <==> f in O(g) and g in O(f)$ - -Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$" - -### The "Dominant Term" Rule -To find which class a function belongs to, simplify it to its core growth rate: -1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$. -2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$. -3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$). - -## Euclidian Algorithm -Purpose is to find the **GCD** (Greatest Common Divisor). - -### Core Rule -Fill out this formula: -$$"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**
| 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/.trash/newfile.md b/.trash/newfile.md deleted file mode 100644 index e69de29..0000000