Overview of Chapters 3.1, 5, 6, and 7.1

Abstract:

Your test will resemble the problems from your homework assignments. You will probably have 10 smaller, equally weighted questions or so (one every 5 minutes!), one of which will involve several true/false questions (like the self-tests at the end of each chapter - answers are at the end of the book).

Section 3.1

• The notation of sets (definition, order, cardinality, empty set, power set, Cartesian products, countable, uncountable, ...)
• Using predicate logic to determine when two sets are equal
• Relationships between sets
• binary and unary operations (and conditions for their proper definition)
• intersection, union, complements, set-difference, and Venn diagrams
• one-to-one correspondence, and proving that cardinalities are the same.

Section 5.1

• Graph definitions and terminology
• Special graphs ( , K(m,n))
• Isomorphic graphs:
  • Definition: Two graphs and are isomorphic if there are bijections (one-to-one and onto mappings) and such that for each arc , (replace braces by parentheses for a directed graph).
  • Theorem: Two simple graphs and are isomorphic if there is a bijection such that for any nodes and of , and are adjacent and are adjacent.
  • Tests for when two graphs are not isomorphic.
• Planer graphs (one which can be drawn in two-dimensions so that its arcs intersect only in nodes)
• Euler's Formula for simple, connected planer graphs states that

  

  where n is the number of nodes, a is the number of arcs, and r is the number of regions (including the infinite region surrounding the graph).

• Computer representations of graphs:
  • the adjacency matrix, and
  • the adjacency list.
  (advantages of one over the other).

Section 5.2

• tree: an acyclic, connected graph with one node designated as the root node (or defined recursively).
• tree terminology
• examples of trees
• tree representations
• tree traversal algorithms:

  

Section 5.3

• decision tree: a tree in which
  • internal nodes represent actions,
  • arcs represent outcomes of an action, and
  • leaves represent final outcomes.
• Examples
• Lower Bounds on Searching
  • 1. Any binary tree of depth d has at most nodes. (Proof: look at the full binary tree, as it has the most nodes per depth.)
    2. Any binary tree with m nodes has depth .
  • Theorem (on the lower bound for searching):

    Any algorithm that solves the search problem for an n-element list by comparing the target element x to the list items must do at least comparisons in the worst case.

• Binary Search Tree (Binary Tree Search)
• Sorting
  • Theorem on the lower bound for sorting: you have to go to at least a depth of in the worst case.

Section 6.2

• Euler Path: a path in which each arc is used exactly once.
• Theorem: in any graph, the number of odd nodes (nodes of odd degree) is even.
• Theorem: an Euler path exists in a connected graph there are either two or zero odd nodes.
• Using the EulerPath algorithm (simply counts up elements in a row i of the matrix (the degree of node i), and checks whether that's even or odd; if in the end there are not zero or two even nodes, there's no Euler path!)
• Hamiltonian Circuit: a cycle using every node of the graph.

Section 6.3

• Shortest Path algorithms (for a simple, positively weighted, connected graph)
  • Dijkstra's Algorithm
  • Bellman-Ford Algorithm
  • Floyd's algorithm
• Minimal Spanning Trees: A spanning tree for a connected graph G is a non-rooted tree containing the nodes of the graph and a subset of the arcs of G. A minimal spanning tree is a spanning tree of least weight of a simple, weighted, connected graph G.
  • Prim's algorithm
  • Kruskal's algorithm

Section 6.4

Traversing a graph (generalizes tree traversal):

• depth-first strategy
• breadth-first strategy

Remember: for the test stick with the convention that, given a choice, we should choose nodes in alphabetic order.

Section 7.1: Boolean Algebra

• How Boolean Algebras generalize propositional logic and set theory.
• The definition, and verifying that is a Boolean algebra
• Additional rules, such as De Morgan's laws, and idempotence.
• Duality
• Proving Boolean algebra equalities