vault backup: 2026-03-09 19:43:13
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Jan Meyer
@@ -29,6 +29,4 @@ Result is the last $"Remainder"$ that is not $0$.
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**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
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1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
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2. **Substitute remainders** -> $$
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2. **Substitute remainders** ->
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284
DAS/merge.md
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284
DAS/merge.md
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## Asymptotic Equivalence Classes (Big-O)
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The equivalence relation definition given in the task is asking you to find functions that grow at the exact same rate (also known as Big-Theta $\Theta$):
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$f asymp g <==> f in O(g) and g in O(f)$
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Notation of $f in O(g)$ means "function $f$ doesn't grow faster than $g$"
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### The "Dominant Term" Rule
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To find which class a function belongs to, simplify it to its core growth rate:
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1. **Drop all lower-order terms:** In $n^3 + n^2$, drop the $n^2$.
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2. **Drop all constant multipliers:** $5n^2$ and $1000n^2$ both become just $n^2$.
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3. **Identify the highest rank:** Factorials ($n!$) > Exponentials ($2^n$, $e^n$) > Polynomials ($n^3$, $n^2$) > Linear ($n$) > Logarithmic ($log n$) > Constant ($1$).
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## Euclidian Algorithm
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Purpose is to find the **GCD** (Greatest Common Divisor).
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### Core Rule
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Fill out this formula:
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$$"Dividend" = ("Quotient" * "Divisor") + "Remainder"$$
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1. **Divide** the bigger number by the smaller one
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2. **How many times** does it fit -> $"Quotient"$
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3. Find out **whats leftover** -> $"Remainder"$
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4. **Shift to left** and repeat ($"Old Divisor" -> "Dividend"$, $"Remainder" -> "Divisor"$)
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Result is the last $"Remainder"$ that is not $0$.
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### Bezout Coefficients
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**Goal:** find a $x$ and $y$ so that $"Divident" * x + "Divisor" * y = gcd("Dividend", "Divisor")$
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1. **Rewrite Euclid (above) equations** to solve for remainder ($"Remainder" = "Old Remainder" - "Dividend" * "Divisor"$)
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2. **Substitute remainders** -> ## Inclusion-Exclusion Principle
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Principle that dictates that when combining / overlapping sets, you have to make sure to not include elements that occur in multiple sets multiple times.
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### Example:
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How many integers between $1$ and $10^6$ are of the form $x^2$ or $x^5$ for some $x in NN$?
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#### How many $x^2$?
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$sqrt(10^6) = 10^3 = 1.000$
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#### How many $x^5$?
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By estimation:
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$$
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&15^5 = 759,375 && "--- in the range / below" 10^6 \
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&16^5 = 1,048,576 && "--- outside the range / above" 10^6 \
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&=> 15 "numbers in the form "x^5"exist" &&
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$$
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> [!warning]
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> Now, we can't add $1,000$ and $15$, since there are numbers that match both, so we need to subtract these duplicates.
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#### How many $x^2$ and $x^5$ / $x^10$?
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$$
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& 3^10 = 59,049 \
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& 4^10 = 1,048,576 \
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& => 3 "numbers that are both" x^2 "and" x^5 "exist"
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$$
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#### Final calculation:
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Formula: $"Elements that are" x^2 + "Elements that are" x^5 - "Elements that are both"$
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$==> 1,000 + 15 - 3 = 1,012$
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### For two sets
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$|"Total possibilities"| - |"Avoid 1"| - |"Avoid 2"| + |"Avoid Both"|$
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Where
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- $"Avoid 1"$ are all elements *not matching* condition 1
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- $"Avoid 2"$ are all elements *not matching* condition 2
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- $"Avoid Both"$ are all elements *not matching* condition 1 **and** 2
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## Type of Task: Boolean Lattice
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Find the amount of upper bounds in ${0, 1}^5$ for the set of vectors $S = {x, y, z}$
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$$
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& x = (0, 1, 0, 0, 0) \
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& y = (0, 0, 1, 0, 1) \
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& z = (0, 1, 1, 0, 0) \
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$$
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> [!INFO]
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> Method used is called **All-Zero Column Method**
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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$.
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To get our result we calculate $2^k$ since there are only two possible states for each value $(0, 1)$:
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$2^2 = 4$.
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So we have **4** upper bounds in total.
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> [!INFO]
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> For the amount of lower bounds we'd check for all-ones columns
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A function takes in an element from its **domain**, transforms it in someway and outputs that transformed element, which is part of the **co-domain**.
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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**.
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A function must be **deterministic** - one input can only map to a single output.
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## Notation
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General function notation: $f: X -> Y$
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> [!INFO]
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> $f$: name of the function
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> $X$: Domain
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> $Y$: Co-domain
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> $f(x)$: Image of $f$
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> $X$, $f(x)$ and $Y$ are [[Set Theory | Set]]
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For any $x in X$ the output $f(x)$ is an element of $Y$.
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## Mapping Properties
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### Injectivity
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A function is _injective_ if every element in $y in f(x)$ has _at most_ one matching $x in X$.
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- $forall y in Y,exists excl x in X : f(x) = y$
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### Surjectivity
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A function is _surjective_ if every element $y in Y$ has _at minimum_ one matching $x in X$
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- $forall y in Y, exists x in X : f(x) = y$
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### Bijectivity
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A function is _bijective_ if every element $y in Y$ has _exactly_ one matching $x in X$ (it is _injective_ and _surjective_)
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- $forall y in Y, exists excl x in X : f(x) = y$## Operators
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| Operation | Explanation | Notation |
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| ----------------- | ------------------------------------ | --------- |
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| **and**<br> | Both $p$ and $q$ must be true | $p and q$ |
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| **or** | Either $p$ or $q$ (or both) are true | $p or q$ |
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| **not** | Negates the statement | $not p$ |
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| **Implication** | If $p$ then $q$ | $=>$ |
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| **Biconditional** | $p$ if and _only_ if $q$ | $<=>$ |
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| **xor** | Either $p$ or $q$ but not both | $xor$ |
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### Implied Operators
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| Operation | Explanion | Notation |
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| --------- | --------------------------------------- | -------------- |
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| **nand** | $p$ and $q$ are not both true | $not(p and q)$ |
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| **nor** | neither of $p$ and $q$ are true | $not(p or q)$ |
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| **xnor** | $p$ and $q$ are both false or both true | $not xor$ |
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## Types of Relations
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| Relation | Explanation | Example |
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| ---------------- | :-------------------------------------------------------------------------------------------------------------------- | ------------------------- |
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| *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$ |
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| *reflexive* | every element is related to itself with the given relation | $a <= a, 5 = 5$ |
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| *anti-reflexive* | every element is *NOT* related to itself in the given relation | $a < a$ |
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| *symmetric* | the given relation work both ways | $a = b => b = a$ |
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| *antisymmetric* | the given relation only works both ways if $a$ and $b$ are the same | $a <= b, b <= a => a = b$ |
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## Equivalence Relations
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A relation $R$ is called _equivalence relation_ when it is _transitive, reflexive and symmetric_.
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### Example:
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**Question:** How many equivalence classes are there for the given equivalence relation?
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$$
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& ~ "on" {0, 1, 2, 3}^(2) \
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& "defined by" (x_1, y_1) ~ (x_2, y_2) <==> x_1 + y_1 = x_2 + y_2
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$$
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> [!INFO]
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> Meaning:
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> 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.
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Solving:
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- Smallest possible sum: $(0 + 0) = 0$
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- Biggest possible sum: $(3 + 3) = 6$
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- All possible sums: $0, 1, 2, 3, 4, 5, 6$
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Each possible sum creates it's own equivalence class. So there are $7$ equivalence classes.
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> [!NOTE]
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> All equivalence classes:
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> $[0]_(~) = {(0, 0)}$
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> $[1]_(~) = {(0, 1), (1, 0)}$
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> $[2]_(~) = {(0, 2), (1, 1), (2, 0)}$
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>$[3]_(~) = {(0, 3), (1, 2), (2, 1), (3, 0)}$
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>$[4]_(~) = {(1, 3), (2, 2), (3, 1)}$
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> $[5]_(~) = {(2, 3), (3, 2)}$
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> $[6]_(~) = {(3, 3)}$
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## Binary Relation
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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$
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## Converse Relation
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$C^top$ or $C^(-1)$ is the relation that occurs if the elements of a _binary relation_ are switched.
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## Composition of Relations - Example
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$$
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"Compute" Q^top compose R "with:"\
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Q = {(2, 2), (3, 3), (2, 1)} \
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R = {(1, 2), (3, 3), (3, 1)} \
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$$
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### 1. Apply converse to $Q$:
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$$
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Q^top = {(2, 2), (3, 3), (1, 2)}
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$$
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### 2. Perform Composition:
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Look at each pair in $R$, check if $Q^top$ has a pair starting with se second element in that pair:
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$$
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(1, 2) -> (2, 2) => (1, 2) \
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(3, 3) -> (3, 3) => (3, 3) \
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(3, 1) -> (1, 2) => (3, 2)
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$$
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### 3. Result:
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$$ Q^top compose R = {(1, 2), (3, 2), (3, 3)} $$
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## Orders
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An **Order** is a mathematical way to sort, rank or compare elements within a set, where some elements come "before" and "after" others.
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A _binary relation_ is called an order if it is...
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- [?] a *reflexive relation*
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- [?] a *antisymmetric relation*
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- [?] a *transitive relation*
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A set is a collection of _unordered_ elements.
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A set cannot contain duplicates.
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## Notation
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### Set Notation
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Declaration of a set $A$ with elements $a$, $b$, $c$:
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$$ A := {a, b, c} $$
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### Cardinality
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Amount of Elements in a set $A$
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Notation: $|A|$
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$$
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A := {1, 2, 3, 4} \
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|A| = 4
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$$
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### Well-Known Sets
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- Empty Set: $emptyset = {}$
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- Natural Numbers: $N = {1, 2, 3, ...}$
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- Integers: $ZZ = {-2, -1, 0, 1, 2}$
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- Rational Numbers: $QQ = {1/2, 22/7 }$
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- Real Numbers: $RR = {1, pi, sqrt(2)}$
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- Complex Numbers: $CC = {i, pi, 1, sqrt(-1)}$
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### Set-Builder Notation
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Common form of notation to create sets without explicitly specifying elements.
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$$
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A := {x in N | 0 <= x <= 5} \
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A = {1, 2, 3, 4, 5}
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$$
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### Member of
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Denote whether $x$ is an element of the set $A$
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Notation: $x in A$
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Negation: $x in.not A$
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### Subsets
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| Type | Explanation | Notation |
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| ------------------------ | ---------------------------------------------------------------------- | ------------------- |
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| **Subset** | Every element of $A$ is in $B$ | $A subset B$ |
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| **Subset or equal to** | Every element of $A$ is in $B$, or they are the exactly same set | $A subset.eq B$ |
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| **Proper subset** | Every element of $A$ is in $B$, but $A$ is definitely smaller than $B$ | $A subset.sq B$<br> |
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| **Superset**<br> | $A$ contains everything that is in $B$ | $A supset B$ |
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| **Superset or equal to** | $A$ contains everything that is in $B$, or they are identical | $A supset.eq B$ |
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## Operations
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### Union
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Notation: $A union B$
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Definition: all elements from both sets _without adding duplicates_
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$$ A := {1, 2, 3}\ B := {3, 4, 5}\ A union B = {1, 2, 3, 4, 5} $$
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### Intersection
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Notation:$A inter B$
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Definition: all elements _contained in both sets_
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$$
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A := {1, 2, 3} \
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B := {2, 3, 4} \
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A inter B = {2, 3}
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$$
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### Difference
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Notation: $A backslash B$
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Definition: all elements _in $A$ that are not in $B$_
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$$
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A := {1, 2, 3} \
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B := {3, 4, 5} \
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A backslash B = {1, 2}
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$$
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### Symmetric Difference
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Notation: $A Delta B$
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Definition: all elements _only in $A$ or only in $B$_
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$$
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A := {1, 2, 3} \
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B := {2, 3, 4} \
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A Delta B = {1, 4}
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$$
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### Cartesian Product
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Notation: $A times B$
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Definition: all pairs of all elements in $A$ and $B$
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$$
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A := {1, 2} \
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B := {3, 4, 5} \
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A times B = {(1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5)}
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$$
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### Powerset
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Notation: $cal(P)(A)$
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Definition: all possible _Subsets of A_
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$$
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A := {1, 2, 3} \
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cal(P)(A) = {emptyset, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
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$$
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