uni_notes/.trash/merge.md

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

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}}