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## Math 109 Winter 2020 Mathematical Reasoning ### Lecture Notes

• Lecture 1  (Mon 01/06/20) Sets, elements, Venn diagrams, cardinality, subsets, the empty set (1.1-1.2).
• Lecture 2  (Wed 01/08/20) Power sets, complement, union, intersection, disjointness, differences, de Morgan (1.2-1.3).
• Lecture 3  (Fri 01/10/20) Union and intersection of an indexed collection of subsets, Boolean properties (1.4, 4.4-4.6).
• Lecture 4  (Mon 01/13/20) Distributive laws etc., partitions, pairwise disjoint vs empty intersection (4.5, 1.5).
• Lecture 5  (Wed 01/15/20) Cartesian products, relations, logic, negation, conjunction/disjunction, truth tables (2.1-2.3, 2.8-2.9).
• Lecture 6  (Fri 01/17/20) Implications, bi-implications, the contrapositive, contradictions, squareroot 2 (2.4-2.7).
• Lecture 7  (Wed 01/22/20) Quantifiers and negation, the absolute value, triangle inequality, sequences, convergence (2.10, 14.1).
• Lecture 8  (Fri 01/24/20) Sequences continued, convergence and divergence, boundedness, sums and products of limits (14.1).
• Lecture 9  (Wed 01/29/20) Partitions and relations continued, equivalence relations, equivalence classes, similar matrices (9.1-9.4).
• Lecture 10  (Fri 01/31/20) Congruent integers, residue classes modulo N, division with remainder, Z_N (9.5, 12.2).
• Lecture 11  (Mon 02/03/20) Division algorithm, well-ordering principle, Z_N has N elements (12.2, 6.1, 9.5).
• Lecture 12  (Wed 02/05/20) Addition and multiplication on Z_N, tables in the example N=3, multiples of 3 and digit-sums (9.6).
• Lecture 13  (Fri 02/07/20) Functions, surjectivity, injectivity, bijectivity, inverse functions (10.1-10.3).
• Lecture 14  (Mon 02/10/20) Composition of functions, a composition of bijections is a bijection (10.4).
• Lecture 15  (Wed 02/12/20) Bijections, inverse function, left/right inverses, image and inverse image, range (10.1, 10.3, 10.5).
• Lecture 16  (Fri 02/14/20) Image and inverse image of unions/intersections, open sets in R, continuity (10.1, 14.5).
• Lecture 17  (Wed 02/19/20) Injections, surjections, and cardinality inequalities, the pigeonhole principle (11.1, 13.3).
• Lecture 18  (Fri 02/21/20) Examples of the use of the pigeonhole principle, numerical equivalence, countable sets (11.1-11.2, 13.3).
• Lecture 19  (Wed 02/26/20) |R|=|(a,b)|, X is never numerically equivalent to P(X), Cantor's diagonal argument, R uncountable (11.1-11.4).
• Lecture 20  (Fri 02/28/20) Subsets of countable sets are countable, Q is countable, unions of countable sets (11.1-11.4).
• Lecture 21  (Mon 03/02/20) Algebraic/transcendental numbers, |(0,1)|=|[0,1]|, Schroder-Bernstein, well-ordering, induction, T_n (11.5, 6.1).
• Lecture 22  (Wed 03/04/20) Examples of induction, geometric sums, Bernoulli's inequality, arithmetic and geometric means (6.1-6.2).
• Lecture 23  (Fri 03/06/20) Strong induction principle -> well-ordering, Fibonacci numbers, golden ratio, Binet's formula (6.3).
• Lecture 24  (Mon 03/09/20) Permutations, factorials, binomial coefficients, Pascal's triangle, binomial expansions (13.4-13.6).
• Lecture 25  (Wed 03/11/20) Prime numbers, prime factorization, infinitely many primes, Euclid bound on p_n, asymptotics of pi(x) (12.1, 12.6).