Sets and Logic

SeLo Resources

• The PList: (Problem List for SeLo has hyperlinks in the Table-of-Contents and the Index.)
• Hall's Marriage (Matching) Lemma
• Optional: Our Primer on Polynomials has further information on Algebraic Numbers, for the Curious Ambitious Student.
• Liouville's Theorem (pdf) on approximating reals by rationals. Also: Hilbert's proof that e is transcendental (pdf).
• The New York Times announces that our (yay!, essayless) Class-C now has all solutions.
• Available is a Practice Class-C. The actual exam is shorter, and easier.
• The nifty thrilling Class-B, in living color! cheered students with its recap of questions from Home-B.
  Enthusiatic students are posting solns left and right! -but especially, right toward our Archive.
  And Now... Class-B with purported solns.

• Home-B, for your cognition.

• Look Pa! All 6 SeLo quizzes (pdf), and Bonus.
• Challenge problem #1:

  Find positive integers T,U,W [W for width] so that, letting S = [T+U] be the sum:
  Rectangles S×W and T×W are (Lmino-)tilable. Yet the U×W rectangle is untilable.

  In class we saw a non-rectangular example. Sam posted an efficient rectanglar example. Also available is a less-efficient rectanglar example, and the same but without grid lines.

• Challenge #2, involving domino tilings.

  Show that evey domino-tiling of 6×6 splits, either vertically or horizontally.

  [Hint: Use PHP together with a parity idea.]

  
  Challenge #3:

  Produce a domino-tiling of 5×6 that has no split, neither vertically nor horizontally.

• Following the below Home-A was the delightful Class-A thrilling Complex aficionados from around the Globe.

• The nifty difty, trusty dusty Home-A.

  For zoid brain-storming, Triangle-paper is conveniently available, as well as Square-paper for generalizing Lmino tiling.

  Computer-generated Lmino tiling appear in Pictures of Lmino Tilings (txt). The pictures start about a third of the way down the file. [Ignore the top of the file; it has notes to me, on how to use the code.]

• Spring2024 SeLo syllabus. Please check exam dates w.r.t religious holidays and ROTC events.
• Last year's Robert Long Essay Competition (RLEC) essay titles.
• Fun, Current: Does Zero = One? (pdf). Here are some  proofs  poofs about which you can post to our Archive.
• Future: An Introduction to Isomorphism, via Gambling.
• A std proof of the Inclusion-Exclusion principle (pdf), together with Candy-Store, Derangement and Stirling-number examples.
• Optional: Practice: Binomials, complex arithmetic.

• Please work-through w: Euclidean algorithm (up through “Extended Euclidean...” but skip the proofs)  and  work-through w: Modular arithmetic (through “Applications”).

  The Euclidean algorithm can be conveniently applied in table-form; I call this form “Lightning Bolt ” because the update-rule looks like a lightning-bolt (used thrice). Please read the Lightning-bolt algorithm (pdf), learning the algorithm, but skipping the proofs.
  Suggestion: Print out on paper (yes, actual paper), the practice sheet for LBolt (pdf) and fill-in the tables.

• The first page of Algorithms in Number Theory (pdf), uses LBolt iteratively to compute the GCD of a list of integers, together with its list of Bézout multipliers. Page 2 uses LBolt to solve linear congruences: Find all x where 33x is mod-114 congruent to 18.
• Optional: Examples of fusing congruences (txt) using LBolt.
  Everybody loves the Euler-Fermat thm. Available is Using EFT to solve 102⁷⁰ + 1 =113= b³⁷ (txt), from Prof. William Stein's book.
• Our Primer on cardinality.
  In addition to proof by raster scan, we can prove that NxN is equinumerous with N via w: Boustrophedon, which can even be pronounced pronounced!
• Optional: Number Theory grab-bag (pdf). (I wrote this for a NT class, so we'll need to skip parts.) Optional: A proof of the Chinese Remainder Theorem (pdf) [CRT], as a ring-isomorphism thm. Proves that Euler phi is a multiplicative fnc. An example of using CRT to count roots of a polynomial.
• A free site, Merge PDF, for merging multiple PDFs into one. (I used this several years ago but have not tested it recently.) There are other free ones on the web as well.
  In case you want it, this free [as of 21Dec2020]. site converts PDFs to PNGs.