Discrete Mathematics I — Logic, Proof, and Structure

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• Discrete versus continuous ~8 min
• Objects, collections, and structures ~8 min
• The language of mathematical notation ~8 min
• Counting as a first discrete question ~8 min
• Discrete structures in computer science ~8 min
• Finite, infinite, and a first look at countability ~8 min
• Definition, theorem, proof: how this course is built ~8 min

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• Propositions and truth values ~8 min
• Negation, conjunction, and disjunction ~10 min
• Truth tables ~10 min
• Exclusive or and other connectives ~8 min
• The conditional statement ~10 min
• Converse, inverse, and contrapositive ~10 min
• The biconditional ~8 min
• Tautologies, contradictions, and contingencies ~8 min
• Logical equivalence ~9 min
• De Morgan's laws ~9 min
• Necessary and sufficient conditions ~9 min
• Translating English into symbolic logic ~10 min
• Valid arguments and rules of inference ~10 min

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• Predicates and truth sets ~8 min
• The universal quantifier ~9 min
• The existential quantifier ~8 min
• Why the domain of discourse matters ~8 min
• Negating quantified statements ~9 min
• Nested quantifiers ~9 min
• Order of quantifiers changes meaning ~9 min
• Translating English into predicate logic ~9 min
• Logic in programming and databases ~8 min

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• What a proof must do ~9 min
• Definitions are the raw material ~8 min
• Direct proof ~10 min
• Proof by contrapositive ~9 min
• Proof by contradiction ~10 min
• Proof by cases ~8 min
• Counterexamples and disproof ~8 min
• Existence proofs ~8 min
• Uniqueness proofs ~8 min
• Constructive versus nonconstructive ~8 min
• Choosing a proof strategy ~9 min
• Common proof errors ~9 min

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• The principle of mathematical induction ~9 min
• Anatomy of an induction proof ~8 min
• Induction and summation formulas ~9 min
• Induction with inequalities ~8 min
• Induction with divisibility ~8 min
• Induction on recursively defined objects ~8 min
• Strong induction ~9 min
• When strong induction is needed ~8 min
• The well-ordering principle ~8 min
• Induction, recursion, and algorithms ~8 min

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• Sets, elements, and set-builder notation ~11 min
• Subsets and proper subsets ~10 min
• Set equality and double inclusion ~9 min
• Union, intersection, difference ~11 min
• Complement and the universal set ~9 min
• Venn diagrams ~10 min
• Set identities ~11 min
• Proving set identities two ways ~11 min
• Cartesian products ~9 min
• Power sets ~9 min
• Cardinality of finite sets ~10 min
• Infinite sets and countability ~12 min

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• Functions between sets ~12 min
• Domain, codomain, range, image ~12 min
• One-to-one functions ~12 min
• Onto functions ~12 min
• Bijections ~12 min
• Function composition ~14 min
• Inverse functions ~14 min
• Floor and ceiling ~12 min
• Images and preimages of sets ~14 min
• Growth of functions: a first look ~12 min

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• Binary relations ~9 min
• Representing relations: matrices and digraphs ~8 min
• Reflexive and irreflexive ~8 min
• Symmetric and antisymmetric ~8 min
• Transitive relations ~8 min
• Combining and composing relations ~9 min
• Equivalence relations ~8 min
• Equivalence classes ~8 min
• Partitions ~8 min
• Partial orders ~8 min
• Hasse diagrams ~9 min

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• Sequences and general terms ~9 min
• Arithmetic and geometric sequences, discretely ~9 min
• Recursive definitions ~9 min
• Recursively defined sets and structures ~9 min
• Recurrence relations ~9 min
• Solving by iteration ~9 min
• Linear homogeneous recurrences ~10 min
• The Fibonacci recurrence ~10 min
• Recursion in algorithms ~9 min

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• The product rule ~9 min
• The sum rule ~8 min
• Counting with the complement ~8 min
• Factorials ~7 min
• Permutations ~9 min
• Combinations ~9 min
• Permutations versus combinations ~8 min
• Binomial coefficients and Pascal's triangle ~9 min
• The binomial theorem, counted ~9 min
• Counting with restrictions ~10 min
• Combinatorial proof ~9 min
• The pigeonhole principle ~9 min

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• What a graph is ~8 min
• Simple graphs, multigraphs, loops ~8 min
• Directed graphs ~8 min
• Graph terminology and special graphs ~9 min
• Degree of a vertex ~8 min
• The handshaking theorem ~9 min
• Walks, paths, and trails ~9 min
• Cycles ~8 min
• Connectedness and components ~8 min
• Subgraphs ~8 min
• Graph isomorphism ~9 min

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• What a tree is ~9 min
• Properties of trees ~9 min
• Rooted trees ~9 min
• Binary trees ~9 min
• Tree traversal ~10 min
• Binary search trees ~10 min
• Spanning trees ~10 min
• Counting with trees ~9 min
• Trees in algorithms and data structures ~9 min

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• What an algorithm is ~12 min
• Reading and writing pseudocode ~13 min
• Correctness and termination ~13 min
• Loop invariants ~14 min
• Proving an algorithm correct ~15 min
• Linear search ~12 min
• Binary search ~14 min
• Bubble and insertion sort ~15 min
• Recursive algorithms ~13 min
• Recursion versus iteration ~12 min

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• Counting operations ~9 min
• Growth rates ~9 min
• Big-O ~10 min
• Proving a Big-O bound ~10 min
• Big-Omega ~9 min
• Big-Theta ~9 min
• Common complexity classes ~9 min
• Polynomial versus exponential ~9 min
• Comparing algorithms ~10 min

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• Sample spaces and events ~12 min
• Counting-based probability ~12 min
• Probability rules ~12 min
• Complementary and inclusive-or events ~12 min
• Conditional probability, revisited ~14 min
• Independence ~12 min
• Bayes' theorem, revisited ~14 min
• Random variables ~12 min
• Expected value, revisited ~12 min
• Linearity of expectation ~12 min