Mohawk Valley Community College s8

MOHAWK VALLEY COMMUNITY COLLEGE UTICA, NEW YORK

COURSE OUTLINE

DISCRETE ALGEBRAIC STRUCTURES

Reviewed and Found Acceptable by Norayne Rosero – 5/01 Revised by Norayne Rosero – 1/02 Reviewed and Found Acceptable by Norayne Rosero – 5/02 Reviewed and Found Acceptable by Norayne Rosero – 5/03 Reviewed and Found Acceptable by Norayne Rosero – 5/04 Reviewed and Found Acceptable by Norayne Rosero – 5/05 Reviewed and Found Acceptable by Norayne Rosero – 5/06 Reviewed and Revised by: Norayne Rosero - 5/07 Reviewed and Revised by: Norayne Rosero - 6/08 Reviewed and Revised by: Norayne Rosero – 5/09 Reviewed and Found Acceptable by Robert Bernstein – 5/10 Reviewed and Found Acceptable by Robert Bernstein – 12/10 Reviewed and Found Acceptable by Russell Penner – 5/12 Reviewed and Found Acceptable by Russell Penner – 5/13 Reviewed and Found Acceptable by Russell Penner – 5/14 Reviewed and Found Acceptable by Russell Penner – 1/16 Course Outline

Title: Discrete Algebraic Structures

Catalog Number: MA275

Credit Hours: 4

Lab. Hours: 0

Prerequisite: MA151 Calculus 1

Catalog Description: This course introduces mathematical systems. Topics include methods of proof, sets, logic, functions, relations, graphs, trees, and algebraic systems. (Fall Semester only)

Course Objectives: 1. To raise the student's level of mathematical sophistication by developing logical thinking and introducing the student to the concept of a proof

2. To help the student develop the ability to construct a proof of a mathematical statement.

3. To serve as an introduction to discrete algebraic structures and mathematical systems

4. To introduce the student to the mathematical topics of graphs and trees which are the bases of some types of data structures used with modern, high- speed computers

5. To provide the student with the opportunity to meet the identified competencies of being able to a) effectively communicate with others and b) demonstrate the ability to interact effectively within a diverse society. General Student Outcomes

1. The student will demonstrate the ability to communicate mathematical results by presenting written solutions and/or proofs to various topics within the course. 2. The student will effectively communicate with others through participation in an oral presentation and/or a written paper dealing with a topic of discrete mathematics. 3. The student will demonstrate the ability to interact effectively within a diverse society by participating in a group (of three or more students) presentation on a topic from the area of discrete mathematics. 4. The student will demonstrate an ability to write proofs using rigorous mathematical reasoning. 5. The student will demonstrate an ability to solve word problems using rigorous mathematical reasoning. 6. The student will be able to state a problem correctly, reason analytically to a solution and interpret the results.

SUNY Learning Outcomes

1. The student will develop well reasoned arguments by demonstrating an ability to write proofs. 2. The student will identify, analyze, and evaluate arguments as they occur in their own and other’s work. 3. The student will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics. 4. The student will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally. 5. The student will demonstrate the ability to employ quantitative methods such as arithmetic, algebra, geometry, or statistics to solve problems. 6. The student will demonstrate the ability to estimate and check mathematical results for reasonableness.

Major Topics to be Covered

1. METHODS OF PROOF

Constructing a logical argument and determining what constitutes an acceptable proof of a mathematical statement

The student will be able to: 1.1 Recognize the premises and conclusion of a given mathematical statement. 1.2 Analyze and construct logical arguments.

1.3 Recognize and use the proof techniques of the principle of mathematical induction, the direct method, and the indirect methods of proof by contradiction and proof by contrapositive 1.4 Construct valid proofs for various mathematical statements involving sets, number systems and formal logic statements. 1.5 Provide a counterexample to disprove a false statement.

2. SETS

Notation and examples, Subsets, Operations on Sets, Venn Diagrams, Laws of Set Theory, Cartesian Products, Power Sets

Goal: To develop the student’s understanding of the set theory.

The student will be able to:

2.1 Define the basic terminology of sets. (set, elements of a set, equal sets, subset, Venn Diagrams, finite sets, infinite sets, cardinality, power set, product set and partitions) 2.2 Demonstrate an understanding of the basic terminology of operations on sets. (union, intersection, disjoint sets, complement, symmetric difference) 2.3 Perform operations on sets using the laws of set theory. 2.4 Apply the addition principle for cardinality of sets.

3. LOGIC

Propositions, Truth Tables and Arguments, Laws of Logic, Quantifiers, Mathematical Induction

3.1 Demonstrate an understanding of the basic terminology dealing with propositional calculus. (premise, conclusion, proposition, propositional variables) 3.2 Create compound logic statements using the connectives of conjunction, disjunction, and implication. 3.3 Write the negation, contrapositive, and converse of compound statements. 3.4 Translate compound statements into symbolic form, given the English statement. 3.5 Determine truth values of compound logic statements using truth tables. 3.6 Present a formal logic proof with steps verified using appropriate rules of inference. 3.7 Translate and determine the truth value of propositional functions involving universal and/or existential quantification of variables. 3.8 Prove the truth of a statement P(n), for n≥no using the principle of mathematical induction.

4. PERMUTATIONS AND COMBINATIONS

4.1 Recognize and apply the multiplication principle of counting. 4.2 Solve application problems dealing with permutations and combinations.

5. RELATIONS

Definitions and Notations, Properties, Equivalence Relations and Partitions, Graphs of Relations, Closure Operations, Partial Order Relations

5.1 Demonstrate an understanding of the basic terminology dealing with relations between two sets and/or on a set. 5.2 Represent and recognize a relation in ordered pair, matrix and/or digraph form. 5.3 Classify a relation as having the following properties; reflexive, irreflexive, symmetric, asymmetric, antisymmetric, and/or transitive. 5.4 Recognize specific properties of relations from set notation, digraph and relation matrix forms. 5.5 Verify and identify characteristics of equivalence relations and the corresponding equivalence classes. 5.6 Verify and identify characteristics of partial orderings. 5.7 Determine closures (reflexive, symmetric, transitive) for given relations. 5.8 Verify and identify characteristics of recurrence relations.

6. FUNCTIONS Definitions and Notations, Properties, Composition, Identity, Inverses, Recursion and Discrete Functions

6.1 Demonstrate an understanding of the basic terminology dealing with functions between two sets and/or on a set. (domain, codomain, range, image, injective, surjective, bijective, inverse) 6.2 Identify and apply special functions (characteristic, floor, ceiling) on given sets. 6.3 Verify that a given function is, or is not, injective, surjective, and/or bijective 6.4 Determine and verify an inverse for a given bijection.

7. GRAPHS

Basic Terminology and Examples, Types of Graphs, Connectivity, Paths and Circuits, Traversals, Applications

7.1 Demonstrate an understanding of the basic terminology dealing with graphs. (vertices, edges, vertex indegree and outdegree, path, circuit, loop, components, simple, connected, complete) 7.2 Demonstrate an understanding of the basic theory and historical development of Eulerian and Hamiltonian graphs. 7.3 Identify Euler paths and/or circuits for given graph 7.4 Identify Hamiltonian paths and/or circuits for given graphs.

8. TREES

Basic Terminology and Examples, Types of Trees, Subtrees, Traversals

8.1 Demonstrate an understanding of the basic terminology dealing with trees. (undirected tree, rooted tree, ordered tree, n-tree, complete n-tree, binary tree, subtree, vertex, level, height, leaves, parent, offspring, siblings, ancestors, descendents) 8.2 Construct label trees. 8.3 Perform tree searching using preorder, inorder and postorder traversals. 8.4 Construct a minimal spanning tree using Prim’s and/or Kruskal’s alogrithms.

9. ALGEBRAIC AND MATHEMATICAL SYSTEMS

Basic Terminology Examples, of Semigroups, Monoids and Groups and Isomorphisms of Groups

9.1 Define and identify selected binary operations on given sets. 9.2 Identify properties of specified binary operations on given sets. 9.3 Identify and verify properties of semigroups, monoids, and groups. 9.4 Identify and verify Isomorphic groups 9.5 Identify and verify properties of Congruence relations 9.6 Identify and verify properties of selected Abelian groups

TEACHING GUIDE

Title: Discrete Algebraic Structures

Catalog Number: MA275

Credit Hours: 4

Lab. Hours: 0

Prerequisite: MA151 Calculus 1

Catalog Description: This course introduces mathematical systems. Topics include methods of proof, sets, logic, functions, relations, graphs, trees, and algebraic systems. (Fall Semester only)

Text: Discrete Mathematical Structures, 6th edition, Kolman, Busby, & Ross, 2009, Prentice-Hall, Inc.

To meet course objective 4, the instructor MUST provide opportunities for the students to a) communicate in written and/or oral fashion and b) to collaborate with each other.

Some suggestions to implement meeting course objective 5:

After chapter five has been covered, the student should have sufficient background knowledge in the importance and usefulness of mathematical reasoning. Included in this process is the ability to communicate mathematical results in a clear and understandable fashion, usually in written form. There are many topics which are not fully treated in the text, but which are of interest. The instructor could have the students form small groups to pursue a common interest. This could involve library research as well as collaborating with others for mutual benefit. The results could be presented in written form, as a paper submitted by the group, or the results could be presented to the class in the form of a "group lecture". Both oral and written reports could be made if the instructor deemed it desirable. Another possibility would be for the students (many of whom are in Computer Science) to present their understanding of ways in which the material in this course is of benefit to them in their major. This, too, could be done collaboratively and presented either orally or in written form. Chapter 1 Fundamentals 4 hours

1.1 Sets and Subsets 1.2 Operations on Sets 1.3 Sequences 1.4 Division in the Integers 1.5 Matrices (Optional) 1.6 Mathematical Structures (Optional)

Chapter 2 Logic 7 hours

2.1 Propositions and Logical Operations 2.2 Conditional Statements 2.3 Methods of Proof 2.4 Mathematical Induction 2.5 Mathematical Statements 2.6 Logic and Problem Solving

Chapter 3 Counting 5 hours

3.1 Permutations 3.2 Combinations 3.3 Pigeonhole Principle (omit) 3.4 Elements of Probability (omit) 3.5 Recurrence Relations

Chapter 4 Relations and Digraphs 7 hours

4.1 Product Sets and Partitions 4.2 Relations and Digraphs 4.3 Paths in Relations and Digraphs 4.4 Properties of Relations 4.5 Equivalence Relations 4.6 Data Structures for Relations and Digraphs (optional) 4.7 Operations on Relations 4.8 Transitive Closure and Warshall's Algorithm

Chapter 5 Functions 5 hours

5.1 Functions 5.2 Functions for Computer Science 5.3 Growth of Functions (omit) 5.4 Permutation Functions (omit) Chapter 6 Order Relations and Structures 2 hours

6.1 Partially Ordered Sets 6.2 Extremal Elements of Partially Ordered Sets (omit) 6.3 Lattices (omit) 6.4 Finite Boolean Algebras (omit) 6.5 Functions on Boolean Algebras (omit) 6.6 Circuit Designs (omit)

Chapter 7 Trees 6 hours

7.1 Trees 7.2 Labeled Trees 7.3 Tree Searching 7.4 Undirected Trees 7.5 Minimal Spanning Trees

Chapter 8 Topics in Graph Theory 5 hours

8.1 Graphs 8.2 Euler Paths and Circuits 8.3 Hamiltonian Paths and Circuits 8.4 Transport Networks (optional) 8.5 Matching Problems (omit) 8.6 Coloring Graphs (optional)

Chapter 9 Semigroups and Groups 8 hours

9.1 Binary Operations Revisited 9.2 Semigroups 9.3 Products and Quotients of Semigroups 9.4 Groups 9.5 Products and Quotients of Groups (optional) 9.6 Other Mathematical Structures (omit)

Note: Care should be taken in assigning reading and homework in Chapter 9 to avoid material that is based on topics that have been omitted.

Presentations: The teaching guide allows for 4 hours for student presentations.

Assessments: The teaching guide allows 4 additional hours for the in-class assessment of student learning. A two hour comprehensive final examination will also be given.