CSC228 Fall 2012 Topics Covered Prof Emile C Chi
Propositional Logic: Truth tables, tautologies, De Morgan's laws
Predicate Logic: Predicates, Quantifiers, Nested Quantifiers, Rules of Inference (Modus ponens et al)
Sets: Operations, subsets, complement, Venn diagrams, De Morgan's laws
Numbers: Natural numbers N , integers Z , rationals Q , reals R
Prime numbers
Counting: 1-1 Correspondance with a subset of N
Methods of Proof: Rules of Inference,
Dirrect: for N : if n is odd then n square is odd if n is even then n square is even
Every rational number can be written as a/b where a and b have no common divisors
Contrapositive: to show p -> q show that ~q -> !p if 3*n + 2 is odd then n is odd if n 2 is odd then n is odd
Contradiction: to show p, show that ~p is a contradiction
There are not a finite number of primes and there is no largest prime
Hard Proofs: the square root of 2 is not a rational number
1 = 2 (ch 1 p3 ppt slide 48)
Fermat's Last Theorem (ch 1 p3 ppt slide 69)
The Four Color Problem
Set cardinality: finite, countable (Hilbert Hotel), uncountable (Cantor Diagonal argument) (ch 2 ppt slides 20-24, 107-117)
Russell's Paradox
Power sets, cardinality of the power set
Cartesian Products, tuples, relations (ch 2 slide 22- error on 24
Functions: Domain, Range, 1-1, onto, bijections, composition, factorlal (ch 2 ppt slides 48-68,74)
Sequences, recurrence relations: Fibonacci sequence (ch 2 ppt slides 79-104)
Series (ch 2 ppt 99-92)
Computability (ch 2 ppt 118, Exercises 38, 39)
Matrices (ch 2 ppt 120-131)
Algorithms (ch 3 ppt 6-10), Greedy Algorithms (ch 3 ppt 21-26), The Halting Problem (Turing) (ch 3 ppt 30-32)
Growth of Functions: Big-O, Big- Ω, Big-Θ (ch 3 ppt 34-57)
Complexity of Algorithms (ch 3 ppt 60-70,78-81), P vs NP, Boolean satisfiabiliy: SAT, 2SAT, 3SAT
Induction (ch 5 ppt slides 1-10,13,15,16), Recursion (33-38, 42-47)
Recursive Algorithms (ch 5 ppt 60-62 ERROR in 62!, 63, 66)
Applications of Recurrence Relations (ch 8 ppt 1-16)
Relations (ch 9 ppt 1-44) Binary relations on a set; Reflexive, Symmetric,Transitive, Equivalence relations: Integers mod m,
Composition, powers, representing using Boolean matrices, digraphs, transitive closure, the connectivity relation
Graphs (ch 10 ppt 1-88) Digraphs, cycles, simple graphs; computer networks, social networks, collaboration graphs: Bacon number, Erdos number; transportation graphs, precedence graphs
Degree of a vertex, handshaking theorem, theorem: a graph must have an even number of odd degree vertices;
Directed graphs: Theorem: the number of edges = the sum of the indegrees = the sum of the outdegrees
Complete graph K n
: has exactly one edge beween any two distince vertices
Cycle C n has one edge from v i to v i+1 a for all i and from v n to v
1 n-dimensional hypercube
Bipartite graphs, complete bipartite graphs, subgraphs
Representing graphs: adjacency list, adjacency matrix
Graph Isomorphism
Paths, Erdos & Bacon Numbers,Connected Components
Euler Paths & Circuits
Hamilton Paths & Circuits
Shortest Path Problems & Dijkstra’s Algorithm (Shortest Path & Dijkstra ppts) Neat Dijkstra App
The Travelling Salesperson Problem
Planar Graphs
Graph Coloring