Unit 6: Set, Number & Graph Theory

Unit 6: Set, Number & Graph Theory Review

Set Theory – Lesson 1.5 • principles and methods used by mathematicians to describe the relationships among sets. • Applications • Internet search engines • Businesses use databases built from set theory to organize large amounts of data

A U B = {x: x is a member of A or x is a member of B}

Venn Diagrams

Complement and Difference of Sets • A’ – the complement of A • The difference of sets A and B • B – A = {x: x is a member of B and x is not a member of A}

Graph Theory – Lessons 3.1-3.2 • Constructing models that describe the relationships that occur among a collection of objects. • Applications: • Determining routes • Minimizing costs • Scheduling

Graph – consists of a finite set of points, called vertices, and lines, called edges, that join pairs of vertices. • A vertex of a graph is odd if it is an endpoint of an odd number of edges of the graph. • A vertex is even if it is an endpoint of an even number of edges.

Euler’s Theorem • A graph can be traced if: • It is connected and • It has either no odd vertices or two odd vertices. • A path in a graph is a series of consecutive edges in which no edge is repeated. • The number of edges in a path is called its length. • Euler path – a path containing all the edges of the graph. • Euler Circuit – an Euler path that begins and ends at the same vertex. • Eulerian graph – a graph with all even vertices contains an Euler circuit.

Fleury’s Algorithm – Graph with all even vertices will have an Euler circuit: Begin at any vertex and travel over consecutive edges according to the following rules: • After you have traveled over an edge, erase it. If all the edges for a particular vertex have been erased, then erase the vertex also. • Travel over an edge that is a bridge only if there is no other alternative. • We can eulerize a graph by duplicating some edges to make an odd vertex even.

The Four Color Problem • Using 4 or fewer colors, can we color the vertices of a graph so that no two vertices of the same edge receive the same color?

Lesson 3.2 The Traveling Salesperson Problem • A Hamiltonpath is a path that passes through all the vertices of a graph exactly once • A Hamilton circuit is a Hamilton path that begins and ends at the same vertex. • In a Hamilton path you do not have to trace every edge as required by an Euler path. • A Complete Graph is one in which every pair of vertices is joined by an edge. Denoted by Kn where n is the number of vertices. • (n-1)! circuits. Because direction doesn’t matter, the distinct circuits are (n-1)!/2.

A weighted graph is a graph where numbers (weights or costs) have been attached to each edge. • In a graph without weights, we define the length of a path as the number of edges in it. • In a weighted graph, the path length/weight is a function of the weights of the edges in the path, usually the sum of those weights.

Finding the Hamilton circuit in Kn with the shortest length. • Brute-Force Algorithm • Nearest Neighbor Algorithm (NNA) • The Best – Edge Algorithm

Number Theory • Number Systems – Lesson 4.1 • Hindu-Arabic – what we use today. • Egyptian • Roman • Chinese

Lesson 4.3 Calculating in Other Bases • Convert from base 10 to another base. • Convert from a given base to base 10. • Adding & Subtracting in other bases • Binary, octal, hexadecimal

Review Problems • 1.5: p.62 #’s 15-18 • 2.1: p.129 #’s 1-6 • 3.1 & 3.2: p.182 #’s 1-10 • 4.1: p.232 #’s 1,2,4,5 • 4.3: p.232 #’s 12-14,16

Unit 6: Set, Number & Graph Theory