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+*  Discrete Math Notes [2024-03-30 Sat 22:51] :journal:cs2233:college:discrete_math:
+
+** Laws of Propositional Logic
+:PROPERTIES:
+:ID: 539a691a-ce6d-4711-84ba-57e74ed42f27
+:END:
+*** Domination Laws
+
+- $p ∧ F ≡ F$
+- $p ∨ ≡ T$
+
+*** Double Negation Law
+
+- $¬¬p ≡ p$
+
+*** Complement Laws
+
+- $p ∧ ¬p ≡ F$
+- $¬T ≡ F$
+- $p ∨ ¬p ≡ T$
+- $¬F ≡ T$
+
+*** De Morgan's Laws
+
+- $¬(p ∨ q) ≡ ¬p ∧ ¬q$
+- $¬(p ∧ q) ≡ ¬p ∨ ¬q$
+
+*** Absorption Laws
+
+- $p ∨ (p ∧ q) ≡ p$
+- $p ∧ (p ∨ q) ≡ q$
+
+*** Idempotent Laws
+
+- $p ∨ p ≡ p$
+- $p ∧ p ≡ p$
+
+*** Associative Laws
+
+Modifying the location of grouping doesn't modify the output among like operators.
+
+- $(p ∨ q) ∨ r ≡ p ∨ (q ∨ r)$
+- $(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)$
+
+*** Commutative Laws
+
+Modifying the location of variables doesn't modify the output around like operators.
+
+- $(p ∨ q) ≡ (q ∨ p)$
+- $(p ∧ q) ≡ (q ∧ p)$
+
+*** Distributive Laws
+
+- $p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)$
+- $p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)$
+
+*** Identity Laws
+
+- $p ∨ F ≡ p$
+- $p ∧ T ≡ T$
+
+*** Conditional Identities
+
+- $p → q ≡ ¬p ∨ q$
+- $p ↔ q ≡ (p → q) ∧ (q → p)$
+
+** Even and Odd Integers
+- An integer $x$ is /*even*/ if there is an integer $k$ such that $x = 2k$
+- An integer $x$ is /*odd*/ if there is an integer $k$ such that $x = 2k + 1$
+
+** Parity
+
+- The /*parity*/ of a number is whether the number is odd or even. If two numbers are both even
+  or both odd, then the two numbers have the /*same parity*/. If one number is odd and the other
+  is even is even, then the two numbers have /*opposite parity*/.
+
+** Rational Numbers
+- A number $r$ is /*rational*/ if there exist integers $x$ and $y$ such that $y ≠ 0$ and
+  $r = x/y$.
+
+** Divides
+
+- An integer $x$ /*divides*/ an integer $y$ if and only if $x ≠ 0$ and $y = kx$, for some
+  integer $k$. The fact that $x$ divides $y$ is denoted $x | y$. If $x$ does not divide $y$, then
+  the fact is denoted $x ∤ y$.
+
+** Prime and Composite Numbers
+- An integer $n$ is /*prime*/ if and only if $n > 1$, and the only positive integers that divide
+  $n$ are $1$ and $n$.
+- An integer $n$ is /*composite*/ if and only if $n > 1$, and there is an integer $m$ such that
+  $1 < m < n$ and $m$ divides $n$.
+
+** Consecutive Integers
+- Two integers are /*consecutive*/ if one of the numbers is equal to $1$ plus the other number.
+  Effectively $n = m + 1$.
+
+** Proof by Contrapositive
+- A /*proof by contrapositive*/ proves a conditional theorem of the form $p → c$ by showing that
+  the contrapositive $¬c → ¬p$ is true. In other words, $¬c$ is assumed to be true and $¬p$ is
+  proven as a result of $¬c$.
+
+** Sets
+
+*** Common Mathematical Sets
+
+| Set                | Symbol | Examples of Elements      |
+|--------------------|--------|---------------------------|
+| Natural Numbers    | ℕ      | 0,1,2,3,...               |
+| Integers           | ℤ      | ...,-2,-1,0,1,2,...       |
+| Rational Numbers   | ℚ      | 0, 1/2, 5.23,-5/3, 7      |
+| Irrational Numbers | ℙ      | π, √2, √5/3, -4/√6        |
+| Real Numbers       | ℝ      | 0, 1/2, 5.23, -5/3, π, √2 |
+
+*** Subset
+- If every element in $A$ is also an element of $B$, then $A$ is a /*subset*/ of $B$, denoted as $A ⊆ B$.
+- If there is an element of $A$ that is not an element of $B$, then $A$ is not a subset of $B$,
+  denoted as $A ⊈ B$.
+- If the universal set is $U$, then for every set $A$: $Ø ⊆ A ⊆ U$
+- If $A ⊆ B$ and there is an element of $B$ that is not an element of $A$ (i.e. $A ≠ B$), then
+  $A$ is a /*proper subset*/ of $B$, denoted as $A ⊂ B$.
+*** Power Set
+- The /*power set*/ of a set $A$, denoted $P(A)$, is the set of all subsets of $A$. For example
+  if $A = {1,2,3}$, then: $P(A) = {Ø,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}$
+**** Cardinality of a Power Set
+- Let $A$ be a finite set of cardinality $n$. Then the cardinality of the power set of $A$ is
+  $2^n$, or $|P(A)| = 2^n$
+
+*** Set Operations
+**** Set Intersection
+
+- Let $A$ and $B$ be sets. The $intersection$ of $A$ and $B$, denoted $A ∩ B$ and read "$A$
+  intersect $B$", is the set of all elements that are elements of both $A$ and $B$.
+
+**** Set Union
+
+- The /*union*/ of two sets, $A$ and $B$, denoted $A ∪ B$ and read "$A$ union $B$", is the set
+  of all elements that are elements of $A$ or $B$. The definition for union uses the inclusive
+  or, meaning that if an element is an element of both $A$ and $B$, then it also and element of
+  $A ∪ B$.
+**** Set Difference
+- The /*difference*/ between two sets $A$ and $B$, denoted $A - B$, is the set of elements that
+  are in $A$ but not in $B$.
+- The difference operation is not commutative since it is not necessarily the case that $A - B = B - A$.
+  Consider that $3 - 2 ≠ 2 - 3$, same applies for the set difference operator.
+
+**** Set Symmetric Difference
+- The /*symmetric difference*/ between two sets, $A$ and $B$, denoted $A ⊕ B$, is the set of
+  elements that are a member of exactly one of $A$ and $B$, but not both. An alternate
+  definition of the symmetric difference operation is: $A ⊕ B = (A - B) ∪ (B - A)$
+
+**** Set Complement
+- The /*complement*/ of a set $A$, denoted $ ̅A$, is the set of all elements in $U$ (the
+  universal set) that are not elements of $A$. An alternate definition of $ ̅A$ is
+  $U - A$.
+
+*** Summary of Set Operations
+| Operation            | Notation | Description                    |
+|----------------------|----------|--------------------------------|
+| Intersection         | $A ∩ B$  | ${x: x ∈ A and x ∈ B}$         |
+| Union                | $A ∪ B$  | ${x: x ∈ A and x ∈ B or both}$ |
+| Difference           | $A - B$  | ${x: x ∈ A and x ∉ B}$         |
+| Symmetric Difference | $A ⊕ B$  | ${x: x ∈ A - B or x ∈ B - A}$  |
+| Complement           | $ ̅A$     | ${x: x ∉ A}$                   |
+
+*** Set Identities
+- A /*set identity*/ is an equation involving sets that is true regardless of the contents of the
+  sets in the expression.
+- Set identities are similar to [[id:539a691a-ce6d-4711-84ba-57e74ed42f27][Laws of Propositional Logic]]
+- The sets $U$ and $Ø$ correspond to the constants true (T) and false (F):
+  - $A ∈ Ø ↔ F$
+  - $A ∈ U ↔ T$
+
+[[./assets/set-identities.png][Table of set identities]]
+
+*** Cartesian Products
+
+- An /*ordered pair*/ of items is written $(x,y)$
+- The first /*entry*/ of the ordered pair $(x,y)$ is $x$ and the second entry is $y$
+- The use of parentheses $()$ for an ordered pair indicates that the order of the entries is
+  significant, unlike sets which use curly braces ${}$
+- Two ordered pairs $(x,y)$ and $(u,w)$ are equal if and only if $x = u$ and $y = w$
+- For two sets, $A$ and $B$, the /*Cartesian product*/ of $A$ and $B$ denoted $A × B$, is the
+  set of all ordered pairs in which the first entry is in $A$ and the second entry is in $B$.
+  That is: $A × B = {(a,b): a ∈ A and b ∈ B}$
+- Since the order of the elements in a pair is significant, $A × B$ will not be the same as
+  $B × A$, unless $A = B$, or either $A$ or $B$ is empty. If $A$ and $B$ are finite sets then
+  $|A × B| = |A| × |B|$
+
+*** Strings
+
+- A sequence of characters is called a /*string*/
+- The set of characters used in a set of strings is called the /*alphabet*/ for the set of
+  strings
+- The /*length*/ of a string is the number of characters in the string
+- A /*binary string*/ is a string whose alphabet is {0,1}
+  - A /*bit*/ is a character in a binary string
+  - A binary string of length $n$ is also called an /*$n$-bit string*/
+  - The set of binary strings of length $n$ is denoted as ${0,1}^n$
+- The /*empty string*/ is the unique string whose length is 0 and is usually denoted by the
+  symbol $λ$. Since ${0,1}^0$ is the set of all binary strings of length 0, ${0,1}^0 = {λ}$
+- If $s$ and $t$ are two strings, then the /*concatenation*/ of $s$ and $t$ (denoted $st$) is
+  the string obtained by putting $s$ and $t$ together
+  - Concatenating any string $x$ with the empty string ($λ$) gives back: $x: xλ = x$
+
+** Functions
+
+_Definition of a function:_
+#+begin_src typst
+A function $f$ that maps elements of a set $X$ to elements of a set $Y$, is a subset of $X × Y$
+such that for every $x ∈ X$, there is exactly one $y ∈ Y$ for which $(x,y) ∈ f$.
+
+$f: X → Y$ is the notation to express the fact that $f$ is a function from $X$ to $Y$. The set
+$X$ is called the _*domain*_ of $f$, and the set $Y$ is the _*target*_ of $f$. An alternate word
+for target that is sometimes used is _*co-domain*_. The fact that $f$ maps $x$ to $y$
+(or $(x,y) ∈ f$) can also be denoted as $f(x) = y$
+#+end_src
+
+- If $f$ maps an element of the domain to zero elements or more than one element of the target,
+  then $f$ is not /*well-defined*/.
+- If $f$ is a function mapping $X$ to $Y$ and $X$ is a finite set, then the function $f$ can be
+  specified by listing the pairs $(x,y)$ in $f$. Alternatively, a function with a finite domain
+  can be expressed graphically as an arrow diagram.
+- In an /*arrow diagram*/ for a function $f$, the elements of the domain $X$ are listed on the
+  left and the elements of the target $Y$ are listed on the right. There is an arrow diagram from
+  $x ∈ X$ to $y ∈ Y$ if and only if $(x,y) ∈ f$. Since $f$ is a function, each $x ∈ X$ has
+  exactly one $y ∈ Y$ such that $(x,y) ∈ f$, which means that in the arrow diagram for a
+  function, there is exactly one arrow pointing out of every element in the domain.
+  - See the following example arrow diagram: [[./assets/arrow-diagram.png]]
+- For function $f: X → Y$, an element $y$ is in the /*range*/ of $f$ if and only if there is an
+  $x ∈ X$ such that $(x,y) ∈ f$.
+  - Expressed in set notation: Range of $f = {y: (x,y) ∈ f, for some x ∈ X}$
+*** Function Equality
+
+- Two functions, $f$ and $g$, are /*equal*/ if $f$ and $g$ have the same domain and target, and
+  $f(x) = g(x)$ for every element $x$ in the domain. The notation $f = g$ is used to denote the
+  fact that functions $f$ and $g$ are equal.
+
+*** The Floor Function
+
+- The /*floor function*/ maps a real number to the nearest integer in the downwards direction
+- The floor function is denoted as: $floor(x) = ⌊x⌋$
+
+*** The Ceiling Function
+- The /*ceiling function*/ rounds a real number to the nearest integer in the upwards direction
+- The ceiling function is denoted as: $ceiling(x) = ⌈x⌉$
+
+*** Properties of Functions
+- A function $f: X → Y$ is /*one-to-one*/ or /*injective*/ if $x_1 ≠ x_2$ implies that
+  $f(x_1) ≠ f(x_2)$. That is, $f$ maps different elements in $X$ to different elements in $Y$.
+- A function $f: X → Y$ is /*onto*/ or /*surjective*/ if the range of $f$ is equal to the target
+  $Y$. That is, for every $y ∈ Y$, there is an $x ∈ X$ such that $f(x) = y$.
+- A function is /*bijective*/ if it is both one-to-one and onto. A bijective function is called a
+ /*bijection*/. A bijection is also called a /*one-to-one correspondence*/.
+
+*** Composition of Functions
+- The process of applying a function to the result of another function is called
+ /*composition*/.
+- Definition of /*function composition*/
+  #+begin_src typst
+  $f$ and $g$ are two function, where $f: X → Y$ and $g: Y → Z$. The composition of $g$ with
+  $f$, denoted $g ○ f$, is the function ($g ○ f): X → Z$, such that for all
+  $x ∈ X, (g ○ f)(x) = g(f(x))$
+  #+end_src
+- Composition is /associative/
+- The /*identity function*/ always maps a set onto itself and maps every element onto itself.
+  - The identity function on $A$, denoted $I_A: A → A$, is defined as $I_A(a) = a$, for all
+    $a ∈ A$
+  - If a function $f$ from $A$ to $B$ has an inverse, then $f$ composed with its inverse is the
+    indentity function. If $f(a) = b$, then $f^-1(b) = a$, and
+    $(f^-1 ○ f)(a) = f^-1(f(a)) = f^-1(b) = a$.
+    - Let $f: A → B$ be a bijection. Then $f^-1 ○ f = I_A$ and $f ○ f^-1 = I_B$
+** Computation
+
+*** Introduction to Algorithms
+
+- An /*algorithm*/ is a step-by-step method for solving a problem. A description of an algorithm
+  specifies the input to the problem, the output to the problem, and the sequence of steps to be
+  followed to obtain the output from the input.
+- A description of an algorithm usually includes
+  - A name for the algorithm
+  - A brief description of the task performed by the algorithm
+  - A description of the input
+  - A description of the output
+  - A sequence of steps to follow
+- Algorithms are often described in /*pseudocode*/, which is a language between written English
+  and a computer language
+- An important type of step in an algorithm is an /*assignment*/, in which a variable is given a
+  variable. The assignment operated is denoted by $:=$, for example to assign $x$ to $y$ you
+  would write it as: $x := y$.
+- The output of an algorithm is specified by a /*return*/ statement.
+  - For example, the statement $Return(x)$ would return the value of $x$ as the output of the
+    algorithm.
+  - A return statement in the middle of an algorithm causes the execution of the algorithm to
+    stop at that point and return the indicated value
+**** The if-statement and the if-else-statement
+- An /*if-statement*/ tests a condition, and executes one or more instructions if the condition
+  evaluates to true. In a single line if-statement, a conditon and instruction appear on the
+  same line.
+- An /*if-else-statement*/ tests a condition, executes one or more instructions if the condition
+  evaluates to true, and executes a different set of instructions in the condition evaluates to
+  false.
+
+**** The for-loop
+- In a /*for-loop*/, a block of instructions is executed a fixed number of times as specified in
+  the first line of the for-loop, which defines an /*index*/, a starting value for the index,
+  and a final value for the index.
+- Each repetition of the block of instructions inside the for-loop is called an
+ /*iteration*/.
+- The index is initially assigned a value equal to the starting value, and is incremented just
+  before the next iteration begins.
+- The final iteration of the for-loop happens when the index reaches the final value.
+
+**** The while-loop
+- A /*while-loop*/ iterates an unknown number of times, ending when a certain condition becomes
+  false.
+*** Asymptotic Growth of Functions
+
+- The /*asymptotic growth*/ of the function $f$ is a measure of how fast the output $f(n)$ grows
+  as the input $n$ grows.
+- Classification of functions using Oh, $Ω$, and $Θ$ notation (called /*asymptotic notation*/)
+  provides a way to concisely characterize the asymptotic growth of a function.
+
+**** O-notation
+- The notation $f = O(g)$ is read "$f$ is /*Oh of*/ $g$". $f = O(g)$ essentially means that the
+  function $f(n)$ is less than or equal to $g(n)$, if constant factors are omitted and small
+  values for $n$ are ignored.
+- The $O$ notation serves as a rough upper bound for functions (disregarding constants and
+  small input values)
+- In the expressions $7n^3$ and $5n^2$, the $7$ and the $5$ are called /*constant factors*/
+  because the values of $7$ and $5$ do not depend on the variable $n$.
+- If $f$ is $O(g)$, then there is a positive number $c$ such that when $f(n)$ and $c × g(n)$ are
+  graphed, the graph of $c × g(n)$ will eventually cross $f(n)$ and remain higher than $f(n)$
+  as $n$ gets large.
+- Definition: /Oh notation/
+  #+begin_src typst
+  Let $f$ and $g$ be functions from $ℤ^+$ to $ℝ^>=$. Then $f = O(g)$ if there are positive real
+  numbers $c$ and $n_0$ such that for any $n ∈ ℤ^+$ such that $n >= n_0$: $f(n) <= c × g(n)$.
+  #+end_src
+- The constants $c$ and $n_0$ in the definition of Oh-notation are said to be a /*witness*/ to
+  the fact that $f = O(g)$.
+
+**** Ω Notation
+- The $Ω$ notation provides a lower bound on the growth of a function
+- $Ω$ notation definition:
+  #+begin_src typst
+  Let $f$ and $g$ be functions from $ℤ^+$ to $ℝ^>=$. Then $f = Ω(g)$ if there are positive real
+  numbers $c$ and $n_0$ such that for any $n ∈ ℤ^+$ such that for $n >= n_0$,
+
+      $f(n) >= c × g(n)$
+
+  The notation $f = Ω(g)$ is read "$f$ is _*Omega*_ of $g$".
+  #+end_src
+
+- Relationship between Oh-notation and $Ω$-notation:
+  #+begin_src typst
+  Let $f$ and $g$ be functions from $ℤ^+$ to $ℝ^>=$. Then $f = Ω(g)$ if and only if $g = O(f)$
+  #+end_src
+
+**** 𝛩 notation and polynomials
+- The $𝛩$ notation indicates that two functions have the same rate of growth
+- $𝛩$-notation definition:
+  #+begin_src typst
+  Let $f$ and $g$ be functions from $ℤ^+$ to $ℝ^>=$.
+
+  $f = 𝛩(g)$ if $f = O(g)$ and $f = Ω(g)$
+
+  The notation $f = 𝛩(g)$ is read "$f$ is *_Theta of_* $g(n)$"
+  #+end_src
+- If $f = 𝛩(g)$, then $f$ is said to be /*order of*/ $g$.
+- /*Lower order*/ terms are those that can be removed from $f$ that does not change the /order/
+  of $f$.
+
+**** Asymptotic growth of polynomials
+#+begin_src typst
+Let $p(n)$ be a degree-$k$ polynomial of the form:
+
+$p(n) = a_kn^k + ... + a_1n + a_0$
+
+in which $a_k > 0$. Then $p(n)$ is $𝛩(n^k)$
+#+end_src
+
+**** Asymptotic growth of logarithm functions with different bases
+
+Let $a$ and $b$ be two real numbers greater than $1$, then $log_a(n) = 𝛩(log_b(n))$
+
+**** Growth rate of common functions in analysis of algorithms
+
+- A function that does not depend on $n$ at all is called a /*constant function*/. $f(n) = 17$
+  is an example of a constant function.
+- Any constant function is $𝛩(1)$.
+- If a function $f(n)$ is $𝛩(n)$, we say that $f(n)$ is a =linear= function.
+- A function $f(n)$ is said to be /*polynomial*/ if $f(n)$ is $𝛩(n^k)$ for some positive real
+  number $k$.
+
+**** Common functions in algorithmic complexity.
+
+The following table of functions are given in increasing order of complexity, with the $m$
+in the third from the bottom being any $m > 3$:
+[[./assets/common-funcs-algo-complexity.png][Table here]]
+
+**** Rules about asymptotic growth
+
+[[./assets/rules-asymptotic-growth-of-functions.png][See here]]
+
+*** Analysis of algorithms
+
+- The amount of resources used by an algorithm is referred to as the algorithm's
+ /*computational complexity*/.
+- The primary resources to optimize are the time the algorithm requires to run
+  (/*time complexity*/) and the amount of memory used (/*space complexity*/).
+- The /*time complexity*/ of an algorithm is defined by a function $f: ℤ^+ → ℤ^+$ such that $f(n)$
+  is the maximum number of atomic operations performed by the algorithm on any input of
+  size $n$. $ℤ^+$ is the set of positive integers.
+- The /*asymptotic time complexity*/ of an algorithm is the rate of asymptotic growth of the
+  algorithm's time complexity function $f(n)$.
+
+**** Worst-case complexity
+- The /*worst-case analysis*/ of an algorithm evaluates the time complexity of the algorithm on
+  the input of a particular size that takes the longest time.
+- The /*worst-case*/ time complexity function $f(n)$ is defined to be the maximum number of
+  atomic operations the algorithm requires, where the maximum is taken over all inputs of size
+  $n$.
+- When proving an /*upper bound*/ on the worst-case complexity of an algorithm (using
+  $O$-notation), the upper bound must apply for every input of size $n$.
+- When proving a /*lower bound*/ for the worst-case complexity of an algorithm (using
+  $Ω$-notation), the lower bound need only apply for at least one possible input of size $n$.
+- /*Average-case analysis*/ evaluates the time complexity of an algorithm by determining the
+  average running time of the algorithm on a random input.