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Discrete Mathematics: Course Outline (MAT 457)

Course Objective

At the end of this course the students will be able to use the vocabulary and symbolic notation of higher mathematics in definitions, theorems and problems. Analyze the logical structure of statements symbolically including the proper use of logical connectives, predicates and quantifiers to think logically. Reason and recognize patterns, make conjectures to use mathematical symbols and discern truth values of arguments and to work with existence, quantification and validation conditions. Understand induction and prove propositions using induction, construct truth tables, prove or disprove a hypothesis. Evaluate the truth of a statement using the principles of logic, explain what a proof is and discern between valid proofs and claim that a proof has been performed but in reality has not. To read a proof of a statement and construct a valid proof using different methods which include direct, proof by cases, indirect, contradiction, contraposition, by example/counterexample, and mathematical induction.

Course Contents

Logic, propositional logic, logical equivalence, predicates & quantifiers, and logical reasoning, sets, basics, set operations, functions: one-to-one, onto, inverse, composition, graphs, integers: greatest common divisor, Euclidean algorithm, sequences and summations, mathematical reasoning, proof strategies, mathematical induction, recursive definitions, structural induction, counting, basic rules, pigeon hall principle, Permutations and combinations, Binomial coefficients and Pascal triangle, probability, discrete probability, expected values and variance, relations, properties, combining relations, closures, equivalence, partial ordering, graphs , directed, undirected graphs.