math 347 lecture notes and review sheets

Fall 2021, section D1H

If you have any questions about your homework scores, please contact the grader (Jason Liu, yichen23 at illinois.edu ) first.

For any questions about the homework itself please try using an appropriate forum on Moodle ( Moodle login help is here )

• lecture 1 (01/25/21) Informal notion of a set, element, membership, union, intersection, subset, complement
  lecture 2 (01/27/21) ordered pairs, relations, equivalence relations, functions
  lecture 3 (01/29/21) Arithmetic of the integers.
  lecture 4 (02/01/21) order and positive cones, ordered rings are integral domains, well-ordering principle
  lecture 5 (02/03/21) induction, division algorithm for integers, unions and intersections, partition of a set
  lecture 6 (02/06/21) equivalence relations and partitions
  lecture 7 (02/08/21) construction of Q from Z. The rationals form a commutative ring with 1 and a field, integers modulo n.
  lecture 8 (02/10/21) injective, surjective and bijective functions, invertability of functions.
  lecture 9 (02/12/21) Pigeonhole principle, finite sets, size of finite sets,
  lecture 10 (02/15/21) powerset P(A) of a set A, Cantor's theorem: there is no onto map from A to P(A) , images and preimages of sets
  lecture 11 (02/19/21) Schroder-Bernstein theorem
  lecture 12 (02/22/21)The set of all sets is not a set, axiom of choice, countable and uncountable sets
  lecture 13 (02/24/21) countable and uncountable sets
  lecture 14 (02/26/21) review of ordered rings, least upper bound.
  (03/01/21) review problems for the first midterm.
  (03/03/21) first midterm exam: in class, usual class time
  lecture 15 (03/05/21) least upper bound property of the reals and its consequences: Archimedian property of R, density of rationals.
  lecture 16 (03/08/21) Density of rationals and irrationals, absolute value and its properties, limits of sequences.
  lecture 17 (03/10/21) Cauchy sequences.
  lecture 18 (03/12/21) subrings, homomorphisms, isomorphisms of rings.
  lecture 19 (03/15/21) Image of a homomorphism is a subring. Cauchy sequences are bounded. Cauchy sequences form a ring.
  lecture 20 (03/17/21) Ideals and the corresponding equivalence relations. Quotient rings. Cauchy sequences converging to zero form an ideal. Reals as the quotient ring of rational Cauchy sequences.
  lecture 21 (03/19/21) R, as constructed, is a field
  lecture 22 (03/22/21) R is an Archimedian ordered field. Cauchy sequences of rational numbers have limits in R.
  lecture 23 (03/26/21) R is has the least upper bound property.
  lecture 24 (03/29/21) Any Cauchy sequence of real numbers has a limit. The field of complex numbers.
  lecture 25 (03/31/21) Convergence and Cauchy sequences in C.
  lecture 26 (04/02/21), e^ix = cos x + sin x, comparison test, uniform convergence, ratio test.
  lecture 27 (04/05/21) ratio test, characteristic of a ring, fields of characteristic 0 contain a copy of the rationals.
  lecture 28 (04/07/21) uniqueness of reals (up to isomorphism), first isomorphism theorem
  lecture 29 (04/09/21) Characteristics of integral domains, polynomial rings.
  lecture 30 (04/12/21) division algorithm for rings of polynomials with coefficients in a field, application to quotients of polynomial rings.
  (04/14/21) review problems for the second midterm
  second midterm exam (04/16/21)
  lecture 31 (04/19/21) vector spaces, linear maps and quotients of polynomial rings
  lecture 32 (04/21/21) norms on vector spaces, metric spaces, Cauchy-Schwarz
  lecture 33 (04/23/21) equivalence of 3 norms on R^n, completeness is the same for two equivalent metrics, open balls.
  lecture 34 (04/26/21) notion of a topology, metrics define a topology, convergence of sequences in a topology, equivalent metrics define same topology.
  lecture 35 (04/28/21) continuity.
  lecture 36 (04/30/21) closed, bounded and compact sets, compact sets in metric spaces are closed and bounded.
  lecture 37 (05/03/21) Proof that closed and bounded subsets of R^n are compact. Heine-Borel.
  lecture 38 (05/38/21) Images of compact sets under continuous maps are compact, continuous real valued functions on compact sets achieve max and min.
  review for the final exam. Please report any typos/issues.
  

Last modified: Mon May 03 11:45:35 CST 2021