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Discrete Structures

Discrete Structures as taught by Ken Berman

Set Theory

Sets

unordered collection (group) of zero or more distinct objects

• set theory (operations about sets)
• S,T,U,A,B … for sets
• List elements in curly braces {a, b, c}
• Sets are equal if and only if they contain the same elements.

• Set Builder

  • \(\{x:P(x)\}\), \(\{x|P(x)\}\).
  • The set of all x such that P(x)
  • {x: x is an integer where x>0 and x<5}

• Set relations

  • \(\forall\) for all
  • \(\rightarrow\) implies
  • \(\leftrightarrow\) if and only if
  • \(\exists\) exists
  • \(\nexists\) does not exist
  • \(\wedge\) and
  • \(\vee\) or
  • \(x\in S\) x in S
  • \(x \notin S\) x not in S

• Empty set

  • \(\varnothing = \{\}\)

• Subset and superset

  • \(S\subseteq T\) subset
  • \(S \supseteq T\) superset
  • \(S\subset T\) proper set
  • \(S \supset T\) proper superset

• Cardinality

  • \(|S|\) Cardinality of S or the number of elements in S

• Universal Set

  • \(U\) is the set containing all other sets (in the problem)
  • universe of discorse

• Union

  • \(\{a,b,c\} \cup \{c,e\} = \{a,b,c,e\}\)

• Intersection

  • \(\{a,b,c\} \cap \{c,e\} = \{c\}\)

• Disjointedness

  \(A \cap B = \varnothing\)

• Set difference

  • \(A - B\)
  • \(\{a,b,c\} - \{c,e\} = \{a, b\}\)
  • Set of all elements in A but not B
  • Compliment with universal set

• Compliment

  • \(\bar A = U - A\)

• Symmetric difference

  • \(A \bigoplus B = A \cup B - A \cap B\)

• Cartesian product

  • \(A \times B = \{(a,b) | a \in A \text{\: and \:} b \in B\}\)
  • \(|A| \times |B| = |A \times B|\)

• Generalized Union and Intersection

  Union or intersection of many sets.

Standard Proof techniques

Disproof by Counterexample

Shows that a conjecture is not true by pointing out an example where the conjecture does not hold.

• No nickels
• 1 quarter + 5 pennies
• 3 dimes
• Greedy method is not appropriate with limited change

Proof by Contradiction

Proof that the opposite cannot be true.

• Square root of 2 is irrational

  • \(\sqrt 2 = a/b\)
  • \(a/b\) is simplified
  • a or b or both must be odd (otherwise could be simplified)
  • \(2 = a^2/b^2\)
  • \(a^2 = 2 ** b^2\)
  • \(a^2\) must be even (2 times any number is even)
  • \(a\) is even as well (odd times odd is odd)
  • \(a = 2 ** k\) where k is a / 2
  • \(2 = (2 ** k)^2/b^2 \rightarrow b^2 = 2k^2\)
  • \(b\) is also odd by this method
  • \(a\) and \(b\) cannot be odd
  • \(\sqrt 2\) cannot be rational

Trees

• set of nodes
• first node is root
• every other node has a "parent" node

Two Trees

• Every node that is not a leaf has 2 child nodes

Binary Trees

• Every node has a maximum of 2 children

Logic

Boolean operators

Negation

Conjunction

Disjunction

Exclusive Or

Implication

Bi-conditional

Normal forms

Disjunctive Normal Form (DNF)

• Take all of the true statements in the table and write a clause for them
• Concatenate all of the true clauses together with a disjunction statement \(\vee\)
• \(\neg f \Leftrightarrow (\neg p \wedge \neg q \wedge \neg r) \vee (\neg p \wedge q \wedge \neg r) \vee ( \neg p \wedge q \wedge r) \vee (p \wedge q \wedge r) \vee (p \wedge q \wedge \neg r) \vee (p \wedge q \wedge r)\)

Conjunctive Normal Form (CNF)

• Negate the DNF form
• \(\neg (\neg f) \Leftrightarrow f\)
• Use demorgans law to distribute

Expression Trees

A binary tree representation of the logical expression

\usetikzlibrary{graphs,graphdrawing, arrows.meta}
\usegdlibrary{trees}
\begin{tikzpicture}
  \graph[binary tree layout, edges={black}]{
    "$(p \wedge q) \vee r$" -- {"$p \wedge q$" -- {"$p$", "$q$"}, "$r$"}

};

\end{tikzpicture}

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Set relations

reflexive

reflexive if, for every element \(a \in A\) we have \(aRa \Rightarrow (a, a) \in R\)

• \( A = \{(a, a): a \in A\}\)

Symmetric

symmetric iff \((x,y) \in R \wedge (y,x) \in R\)

Transitive

Iff R relates \(a\) to \(b\) and \(b\) to \( c\) then \(a \) relates to \(c\)

• \(a < b < c \rightarrow a < c\)
• \(a = b = c \rightarrow a = c\)

Modular arithmetic

• \(x \equiv y (\text{mod} \: n) \leftrightarrow (x-y) \: \text {mod} \: n = 0\)

Addition Tables

• Z mod 4

  +	0	1	2	3
  0	\((0 + 0) \mod 4 = 0\)	1	2	3
  1	\((1 + 0) \mod 4 = 1\)	2	3	0
  2	\((2 + 0) \mod 4 = 1\)	3	0	1
  3	\((3 + 0) \mod 4 = 3\)	0	1	2

Multiplication tables

• Z mod 4

  x	0	1	2	3
  0	\((0 \cdot 0) \mod 4 = 0\)	0	0	0
  1	\((1 \cdot 0) \mod 4 = 0\)	1	2	3
  2	\((2 \cdot 0) \mod 4 = 0\)	2	0	2
  3	\((3 \cdot 0) \mod 4 = 0\)	3	2	1

Exam 1 review

[All-Slides](pdfs/combine.pdf)❌

Set Theory

• Union

  • \(S = A \cup B\)

  \(A\)	\(B\)	\(A \cup B\)
  T	T	T
  T	F	T
  F	T	T
  F	F	F

• Intersection

  • \(S = A \cap B\)

  \(A\)	\(B\)	\(A \cup B\)
  T	T	T
  T	F	F
  F	T	F
  F	F	F

• Difference

  • \(S = A - B\)

  \(A\)	\(B\)	\(A \cup B\)
  T	T	T
  T	F	T
  F	T	T
  F	F	F

• Symmetric difference

  • \(S = A \bigoplus B\)
  • \((a \in S \iff (a \in A \quad \text{and} \quad a \ni B)\)

  \(A\)	\(B\)	\(A \cup B\)
  T	T	F
  T	F	T
  F	T	T
  F	F	F

• Demorgans law

  \(\neg (A \cup B) = \neg A \cap \neg B\)

• Principle of Inclusion-Exclusion

  \(|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|\)

Proof Techniques

• Counterexample
• Contradiction
• Induction
• Trees

• Trees

  • n nodes
  • n-1 edges
  • leaf nodes = intermediate nodes + 1
  • Total nodes = intermediate nodes + leaf nodes

power sets

• \(A= \{a, b, c\}\)
• \(P(A) = \varnothing , \{a\}, \{b\}, \{c\}, \{a, b\}, \{a, c\}, \{b, c\}, \{a, b, c\}\)
• \(|P(A)| = 2^{|A|} = 2^3 = 8\)

Propositional logic

• All F = contradiction
• All T = Tautology
• CNF conjunction of all disjunction clauses, unsatisfiable when all combinations of clauses are present
• DNF disjunction of all conjunction clauses
• Logic

• NP and NP-completeness

  • P = problem that can be solved in polynomial time
  • NP = non-deterministic polynomial (unknown if it can be solved in polynomial time)
  • NP-complete = any NP problem A can be reduced to problem B

Functions and relations

• One to one -> (injective)
• Onto () -> surjective
• One to one and Onto -> Bijective
• Density
• Equivalence relations
  • Reflexive, \(a, a \in R \: \text{for every a in A}\)
  • Symmetric, \((b, a \in R\: \text{ whenver} \: a, b \in R\)
  • Transitive, \((a, b) \in R \text{ and } (b, c) \in R \text{ then } (a, c) \in R \text{ where } a, b, c \in A \)
• Asymmetric, \((a, b) \in R \text{ implies } (b, a) \not\in R\)
• AntiSymmetric, assymetric except for the case \((a, b) \in R \rightarrow (b, a) \in R\) where \(b\) is equal to \(a\)
• Poset (partially ordered set)
  • reflexive
  • Antisymmetric
  • Transitive

Mod Arithmetic

• \((x + y) \mod k = (x \mod k \quad + \quad y \mod k) \mod k \)
• \(b^{n-1} = 1 \mod n \)

Exam 2 review

• [All-Slides](pdfs/combine2.pdf)❌ after the first exam.
• [All Topics](pdfs/Topic Coverage for Test 2 CS2071 Fall 2021-1.pdf)❌ most formulas are in this one.

RSA Public Key Cryptosystem

• Extended GCD to compute private key

  • \(\varphi(n) = (p-1)(q-1)\)
  • \(se + t\varphi(n) = g = 1 = gcd(e, \varphi(n))\)
  • \(se \equiv 1(\mod \varphi (n))\)
  • \(s = e^{-1}(\mod \varphi(n))\)

  • R implementation of GCD

    This is an implementation of Eculid's recursive GCD algorithm. Should be easy to convert to python.

    euclid <- function(a, b) {
      print(c(a, b))
      if (b == 0) {
        return(a)
      }
      euclid(b, a %% b)
    }
    

Intro to Graph Theory, Euler's Degree Formula

• A Graph is a series of vertices (nodes) that are connected by edges
• Degree (in this class) is equal to the number of edges that a node is connected to
• Complete graph is a graph where every node is connected to every other node.
• A subgraph is a graph made from a subset of nodes in another graph
• An induced subgraph must have the same edges that the parent graph had.

Graph Isomorphism, Path, Coloring

• Isomorphic graphs are identical except for node position, connections are the same
• nodes in colored Graphs are colored to be different than all of the adjacent nodes.
• A path is a sequence of vertices connected by edges within a graph. Vertices may be repeated. A path is the same as a trail.
• Simple paths are paths where vertices are not repeated.

Planar Graphs and Euler's Polyhedron Formula

• [Supplemental Notes For Planar Graph (Kuratowski)](pdfs/SupplementalNotesPlanarGraphs.pdf)❌
• Planar graphs are graphs that can be represented isomorphically without any overlapping edges.
• \(\sum_{g \in F}\deg(g) = 2m\) where g is a vertex in face F, and m is the number of edges
• 5 regular polyhedra
  • Tetrahedron
  • Cube
  • Dodecahedron
  • Icosahedron
  • Octahedron

Spanning Trees and Eulerian Circuits

• Eulerian path contains all edges in a graph exactly once
• Eulerian circuit is a circuit that contains all edges exactly once.
• Simple path that contains every vertex in the graph is a Hamiltonian Path
• Hamiltonian cycle is a cycle that contains every vertex in the path

Hypercubes and hamiltonian Cycles

Implementation of Graphs and Digraphs

Digraphs

The Web Digraph and PageRank

Intro to Combinatorics and Counting

Permutations and Combinations

Identities, Binomial Theorem, Pascals Triangle

Exam 3 review

[Link to all Slides](pdfs/combine3.pdf)❌