Course Title:  __MTH 288_Discrete Mathematics_____________________

 Course Description

Presents topics in sets, counting, graphs, logic, proofs, functions, relations, mathematical induction, Boolean Algebra, and recurrence relations.

 General Course Purpose

The goal is to give the student a solid grasp of the methods and applications of discrete mathematics to prepare the student for higher level study in mathematics, engineering, computer science, and the sciences.

 Course Prerequisites/Corequisites

Prerequisite:  Completion of MTH 263 Calculus I with a grade of C or better or equivalent.

 Course Objectives

Upon completing the course, the student will be able to:

 Note:  Methods of proofs and applications of proofs are emphasized throughout the course.

Logic - Propositional Calculus

●     Use statements, variables, and logical connectives to translate between English and formal logic.

●     Use a truth table to prove the logical equivalence of statements.

●     Identify conditional statements and their variations.

●     Identify common argument forms.

●     Use truth tables to prove the validity of arguments.

Logic - Predicate Calculus

●     Use predicates and quantifiers to translate between English and formal logic.

●     Use Euler diagrams to prove the validity of arguments with quantifiers.

Logic - Proofs

●     Construct proofs of mathematical statements -  including number theoretic statements - using counter-examples, direct arguments, division into cases, and indirect arguments.

●     Use mathematical induction to prove propositions over the positive integers.

Set Theory

●     Exhibit proper use of set notation, abbreviations for common sets, Cartesian products, and ordered n-tuples.

●     Combine sets using set operations. 

●     List the elements of a power set. 

●     Lists the elements of a cross product.

●     Draw Venn diagrams that represent set operations and set relations.

●     Apply concepts of sets or Venn Diagrams to prove the equality or inequality of infinite or finite sets.

●     Create bijective mappings to prove that two sets do or do not have the same cardinality.

Functions and Relations

●     Identify a function's rule, domain, codomain, and range.

●     Draw and interpret arrow diagrams. 

●     Prove that a function is well-defined, one-to-one, or onto.

●     Given a binary relation on a set, determine if two elements of the set are related.

●     Prove that a relation is an equivalence relation and determine its equivalence classes.

●     Determine if a relation is a partial ordering.

Counting Theory

●     Use the multiplication rule, permutations, combinations, and the pigeonhole principle to count the number of elements in a set.

●     Apply the Binomial Theorem to counting problems.

Graph Theory

●     Identify the features of a graph using definitions and proper graph terminology.

●     Prove statements using the Handshake Theorem.

●     Prove that a graph has an Euler circuit.

●     Identify a minimum spanning tree.

Boolean Algebra

●     Define Boolean Algebra.

●     Apply its concepts to other areas of discrete math.

●     Apply partial orderings to Boolean algebra.

Recurrence Relations

●     Give explicit and recursive descriptions of sequences.

●     Solve recurrence relations.

Major Topics to be Included

Logic – Propositional Calculus

Logic - Predicate Calculus

Logic - Proofs

Set Theory

Functions and Relations

Counting Theory

Graph Theory

Boolean Algebra

Recurrence Relations