About 60% of the Final will be over the new material since the last test, 20% from the test 1 material, and 20% from the test 2 material. Below are the hightlights of the textbook sections since the last exam:

Section 4.1

Definition of a binary relation. Operations on binary relations on a set include union, intersection, and complementation. Properties of binary relations (reflexivity, symmetry, transitivity, and antisymmetry).

Skills:

Be able to test an ordered pair for membership in a binary relation.

Be able to test a binary relation for reflexivity, symmetry, transitivity, and antisymmetry.

Draw the Hasse diagram for a partially ordered set.

Section 4.4 (pp. 289-297; 307-311)

Topics: function, domain, codomain, image, preimage, range, onto/surjective, one-to-one/injective, bijection

Definition of Order of Magnitude (theta notation, ). Definition of Big-Oh

(O( )) notation.

Skills:

Be able to test whether a given relation is a function.

Be able to test whether a function is one-to-one or onto.

Be able to apply the definitions of Order of Magnitude (theta notation, ) and Big-Oh

(O( )) notation to approximate the constant (c) on the fastest growing term to estimate the execution time for a larger problem size (say n=10,000) given a the execution time for a smaller problem size (n=1,000).

For a given section of code containing nested loops, be able to

• provide a reasonable Big-Oh notation (upper-bound estimate on the execution time)
• trace the code to determine how many times the inner-most loop executes
• determine the Order of Magnitude/Theta notation ("exact" estimate on the execution time)

Section 5.1

Definition of a graph as an ordered triple (N, A, g)

Terminology: nodes/vertices, arcs/edges, undirected, directed, weighted, labeled, adjacent nodes, loop, parallel arcs, simple graph, connected, isolated node, reachable, degree of a node, in-degree, out-degree, path, cycle, acyclic, complete graph, subgraph

Computer Representation of a graph: adjacency matrix and adjacency list

Section 5.2

Definition of a tree

Terminology: child, parent, leaf, internal node, forest, depth, height, binary tree, left child, right child, binary search tree, full binary tree, complete tree, expression tree

Representation of a binary tree on the computer: linked/pointer representation, array representation

Proving properties of binary trees using proof by induction: e.g., Prove that a binary tree with n nodes has n-1 arcs.

Section 5.3

General decision tree argument that the best searching will require in the worst-case.

General decision tree argument that the best sorting will require in the worst-case.