SPRINGFIELD TECHNICAL COMMUNITY COLLEGE
School of Mathematics, Sciences, and Engineering Transfer
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Discrete Structures
Math 376
A. Course Description
This course is a study of the discrete structures of mathematics. They include
propositional calculus, quantification, sets, functions, sequences and series, number
theoretic functions, proofs - direct and indirect, induction, combinatorics, discrete
probability, recurrence relations, equivalence relations, partial orderings, graphs, paths,
and finite state machines.
B. Textbooks
Required:
Discrete Mathematics and its Applications, 7th Edition, Kenneth
Rosen, McGraw-Hill
C. Instructional Objectives
Understand and employ the basics of LOGIC.
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Define and construct propositions using unary and binary connectives.
Set up truth tables to establish tautologies, contingencies and contradictions.
Employ bit operators to manipulate string and numeric data.
Translate and symbolize universal and existential quantifiers.
Define and apply FUNCTIONS .
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Define basic set notation terminology and perform set operations.
Employ functions to examine relations between domains and ranges.
Determine the convergence and divergence of sequences and series.
Use exponential and factorial functions to determine function growth.
Employ the concepts and theorems of NUMBER THEORY .
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Determine the computational complexity of an algorithm.2
Use prime and composite numbers in basic number theoretic algorithms.
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Use the division algorithm.
Find the GCD and LCM.
Use the notation and operations of modular arithmetic.
Determine random number seeds and pseudorandomness.
Develop and analyze encryption models.
Perform binary operations.
Write computer code for Euclidean algorithm.
Solve problems using the Chinese Remainder Theorem.
Define basic matrix operations.
Apply MATHEMATICAL REASONING.
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Define axioms and prove basic number theoretic theorems.
Use the rules of inference to determine the validity of arguments.
Recognize fallacies.
Use the direct proof and indirect to prove theorems.
Use mathematical induction to prove theorems for all n.
Demonstrate existence proofs.
Recognize the Halting problem and its consequences.
Use the well-ordering principle.
Set up recursion definitions.
Employ loops and iterations to solve recursion problems.
Determine program correctness.
Calculate probabilities and values using COMBINATORICS.
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State and use the sum rule and product rule for counting.
Use the inclusion/exclusion principles in counting.
Draw tree diagrams for all possible outcomes.
Use Pascal's identity for counting problems.
State Vandermonde's Identity.
Expand binomials with the binomial theorem.
Solve discrete probability problems with combinations and permutations.
Do problems with conditional probability.
Find the expected value of an experiment.
Use Chebychev's inequality for appropriate discrete probability problems.
Compute the complexity of computations.
Examine and compare RELATIONS.
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Set up mathematical models for compound interest, Fibonacci numbers and Tower of
Hanoi problems.
Solve homogeneous linear recurrence relations.
Define binary relations.
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Define and recognize reflexive, symmetric and transitive properties.
Define n-ary relations.
Set up and solve problems using directed graphs.
Define closures and paths in a given path problem.
Define equivalence relations and partial orderings on a set.
Examine and solve problems with GRAPHS.
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Define a simple graph.
Define a multigraph.
Set up graph models to represent real world problems.
Use matrices to represent graphs.
Set up an adjoining matrix and an incidence matrix.
Determine connectivity of a graph.
Find Euler paths of a given graph.
Find Hamilton paths of a given graph.
D. Teaching Procedures
E. Course Topics
Chapter 1: The Foundations: Logic and Proofs
1.1
1.2
1.3
1.4
1.6
1.7
Propositional Logic
Applications of Propositional Logic
Propositional Equivalences
Predicates and Quantifiers
Rules of Inference
Introduction to Proofs
Chapter 2: Basic Structures: Sets, Functions, Sequences, Sums and Matrices
2.1
2.2
2.3
2.4
2.5
2.6
Sets
Set Operations
Functions
Sequences and Summations
Cardinality of Sets
Matrices
Chapter 4: Number Theory and Cryptography
4.1
4.2
4.3
4.6
Divisibility and Modular Arithmetic
Integer Representation and Algorithms
Primes and Greatest Common Divisors
Cryptography
Chapter 6: Counting
6.1
6.2
6.3
6.4
The Basics of Counting
The Pigeonhole Principle
Permutations and Combinations
Generalized Permutations and Combinations
Chapter 7: Discrete Probability
7.1 An Introduction to Discrete Probability
7.2 Probability Theory
7.3 Bayes’ Theorem
Chapter 8: Advanced Counting Techniques
8.1 Applications of Recurrence Relations
8.2 Solving Linear Recurrence Relations
Chapter 9: Relations and Their Properties
9.1 Relations and Their Properties
9.2 n–ary Relations and Their Applications
Chapter 10: Graphs
10.1 Graphs and Graph Models
10.2 Graph Terminology and Special Types of Graphs
10.5 Euler and Hamilton Paths
F. Tentative Test Schedule
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Week 13
Chapter 1
Chapter 1
Test Chapter 1, Chapter 2
Chapter 2
Test Chapter 2, Chapter 4
Chapter 4
Test Chapter 4
Chapter 6
Test Chapter 6
Chapter 7
Chapter 7, Chapter 8
Test Chapters 7 and 8
Chapter 9
Week 14 Test Chapter 9, Chapter 10
Week 15 Test Chapter 10, Review for Final Exam
G. Grading
H. Attendance and Classroom Conduct Policy
I. Office Hours