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Nydus Open Learning

Mathematics

Mathematics · Avanzado · Nivel 1

Introduction to proof and discrete mathematics

Propositional and predicate logic: the connectives, the converse and the contrapositive, logical equivalence, quantifiers and how to negate them. The proof techniques: direct, contrapositive, contradiction, cases and induction. Sets, power sets and products; relations, equivalence relations and partitions; injections, surjections and bijections. Counting with the product and sum rules, permutations and combinations, binomial coefficients and the pigeonhole principle. Graphs: degrees and the handshake lemma, trees, Euler circuits, planarity and bipartite graphs. Number theory: divisibility and factorisation, the Euclidean algorithm and Bézout's identity, and modular arithmetic. And recurrence relations, solved by running them forward and by their characteristic equations.

Ages 18+ 28 lessons about 11 hours in all

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  1. 1 Statements, connectives and truth tables
  2. 2 Implication, converse and contrapositive
  3. 3 Logical equivalence and De Morgan's laws
  4. 4 Quantifiers and the order they come in
  5. 5 Negating a quantified statement
  6. 6 What a proof must do, and the direct one
  7. 7 Proof by contrapositive
  8. 8 Proof by contradiction
  9. 9 Proof by cases, and the single counterexample
  10. 10 Mathematical induction
  11. 11 Sets and the operations on them
  12. 12 Power sets and Cartesian products
  13. 13 Relations and their properties
  14. 14 Equivalence relations and partitions
  15. 15 Injections, surjections and bijections
  16. 16 The product rule and the sum rule
  17. 17 Permutations and combinations
  18. 18 Binomial coefficients and Pascal's rule
  19. 19 The pigeonhole principle
  20. 20 Graphs, degrees and the handshake lemma
  21. 21 Trees and connectivity
  22. 22 Euler circuits and Euler paths
  23. 23 Planar and bipartite graphs
  24. 24 Divisibility, primes and factorisation
  25. 25 The Euclidean algorithm and Bezout's identity
  26. 26 Modular arithmetic
  27. 27 Recurrence relations and closed forms
  28. 28 Linear recurrences and characteristic equations