### 3110 Highlights

• Mon 1/10 Intro to Proofs - Minesweeper rules and proofs, set theoretic definitions and proofs. Class proofs of
3x3 minesweeper with A2=2, C1=1=C3, and C2=*
A AND B SUBSET A
HW for Wed Minesweeper proofs that B3 is a number, B2=3 and B3=2, Read p. 1-13 carefully in the book ( Intro to Algebraic Structures by Landon), and rewrite the proofs of Thm 1,2, and 4 i) so that they are in your own words and they satisfy the checklist points and the proof-writing sheet.
• Wed 1/13 Have students present HW from Mon, look at
U={0,1,2,3,4,5,6,7,8,9,10},
A={0,1,2,3,4,5}
B={0,2,4,6,8,10}
C={2,3,5,7}
Each student is assigned one of A U B, A AND B, A' U B, A AND B AND C, A' AND B AND C, A U(B AND C), A AND (B U C), (A U B')', A \ B, B \ A, A \ (B \ C), C \ (B \ A), (A \ B) AND (C \ B), (A \ B) AND (A \ C), U' AND (A U B U C). HW for Fri Finish up Thm 4i (right side contained in the left side), prove that the answer to your statement is the answer you came up with in class, read p. 13-15 in the book and do p. 15 #3.
• Fri 1/15 Go over p. 15 # 3, Prove or Discprove A and B are sets --> A c A AND B,
Minesweeper 6*6 with A1=1, A2=2, A3=2, A4=1, B4=1, C4=1. Prove that there is a mine in square B2, Intro to Computer Lab and course web pages, Web Search for Bios. presentations of proof of statement assigned in Wednesday's class. HW for Fri 22 326 Walker Problem Set 1, HW for Mon 25 Re-read p. 1-15 in the book, p. 13 #6, 6*6 Minesweeper game from class - what other statements can you prove about this game? Prove them. HW for Wed 27Summary of Life and Work on mathematician due (we'll focus on the listed math aspect in the future - not for this part of the paper).
• MLKJ Week (No class due to MLKJ Day and Conference) Work on HW from Fri 15.
• Mon, Jan 24Give hints on PS1 (revs due Fri), go over definitions of power set of a set, arbitrary intersection and union of sets. Proof of p. 13 #6. HW for Wed Bio paper due, p. 16 #1, 6*6 minesweeper hw from last time, finish up proof of p. 13 #6 - Assume that A U B =E and prove that E \ B c A.
• Wed, Jan 26 Use index cards to choose students to present HW from last time, discuss 6*6 minesweeper. Define AxB and highlight the difference between it and A AND B, define Relation, Range, Domain, and Inverse of a Relation. HW for FriPS 1 revs, p. 18 #3, p. 19#2.
• Fri, Jan 28 Go over p. 18#3 and p. 19 2a. Define function. HW for Mon p. 19 2 b,c,d, p. 20 3,4, finish proof from class.
• Mon, Jan 31Go over hw, go over function notations and definitions. Hand out PS 2 due next Mon.HW for Wedp. 19 2d, p. 20 3 (I(-1) Subset I), p. 20 4, p. 22 1,5.
• Wed, Feb 3Go over hw, go over 1-1 functions. Disprove that f(x)=x^2 is injective. HW for Frip. 20 #4 1-1, p. 20 #5.
• Fri, Feb 5Students go over hw, Prove that f(x)=x^2 is injective, review defs of function, and 1-1 and discuss proofs (and disproofs). Def of onto. Proof that f(x)=(x+1,x^2) is not onto. HW for Mon Is p. 20 #4 an onto function? Is f(x)=x^2 onto? EC part 2 of function proof for p. 20#4
• Mon, Feb 7Students go over hw. Hand out PS 3, and proof-writing continued sheet. Go over geometric meaning of function, 1-1 and onto. Give examples of 1-1 but not onto function, and onto but not 1-1 function. Go over definition of composite functions and p. 24#1. Definition game for A subset B. I say something true about this, next person repeats what I say and adds something,... HW for WedLook at f(x)=x^2 and g(x)=x+1, What is f o g and g o f? Are the bijections? Prove or Disprove.
• Wed, Feb 9 Students present hw from Mon, Introduction to Number Theory and a Lemma needed to prove that sqrt(2) is irrational. HWWork on PS 3 due Mon, PS 2 revs due 1 week from tomorrow, and reading due in 1 week (read p. 28-51 and take notes in your own words). Lagrange, Abel and Germain presentations next week.
• Fri, Feb 11 Proof that sqrt(2) is irrational, disprove that xsqrt(2) is irrational for all x in R\{0}. HW for Mon Prove x sqrt(2) is irrational for all x in Q\{0}, PS 3 due Mon at 5, Listen to Twins Sherri and Terri jumping rope- Cross my heart and hope to die. Here's the digits that make pi- 3.1415926535897932384... in Simpsons episode 3G02 "Lisa's Sax" 10/19/97
• Mon Feb 14 Students work in groups of 2 on a proof which they put up on the board.
a,b in C implies ab in C
The square of any integer is in the form 4k or 4k+1
Any integer of the form 6k+5 is of the form 3k+2, but not conversly
Disprove there exists y in R\{0} s.t. for all x in R\{0} xy=1
Prove For all x in R\{0} there exists y in R\{0} s.t. xy=1
Prove For all x,y in R, x not equal to y IMPLIES 2x+1 not equal to 2y+1 by stating and proving the contrapositive.
Hand out Problem Set 4 due Tues 22nd at 5pm
• Wed Feb 16 Go over HW that x sqrt(2) is irrational for all x in Q\{0}. Discuss book readings on reals, rationals, irrationals, natural numbers and complex numbers. Students put their questions on the board.
• Fri Feb 18 Amanda's Presentation on Lagrange and the Solution of 3rd and 4th degree polynomials by radicals and Dr. Sarah's Modular Arithmetic Maple Demo HW for Monday Review Maple Demo.
• Mon Feb 21 Alan's Presentation on Abel and Solving a quintic equation, Review of Maple Demo
• Wed Feb 23Lizette's presentation on Germain and and Fermat's Last Theorem and Sophie Germain Numbers and videos
Star Trek TNG The Royale Episode 38 3/27/89, stardate 42625.4 part where Captain Pickard and Riker discuss Fermat's Last Theorem.
The Proof, NOVA, 1997 about Fermat's Last Theorem.
• Fri, Feb 25 Dr. Sarah's web page and Maple demo on the solution of the quadratic, cubic, quartic and quintic by radicals
• Mon, Feb 28 Finish The Proof Video, review ideas in the video such as the number of Natural Number solutions of x^2+y^2=z^2, the statement of Fermat, pictures of Elliptic curves and Modular Forms, The statement of Taniyama Shimura, the epsilon conjecture and why Taniyama Shimura implies Fermat. HW for Wed review Friday's Maple Demo.
• Wed, March 1Review Friday's Maple Demo, start proof by induction.
• Fri, March 3Test 1
• Mon, March 6Induction proof that 3/(5^n-2^n) for all n, Proof by contradiction that Induction works, using the least natural number principle, work in groups of 2 on presentations for Wed. HW for WedReview Negations from Solutions for PS 4, work on presentations for Wed, work on PS 5 due Fri.
• Wed March 8Definition of a group. Proof that Q+ is a group under multiplication, but that Q is not a group under multiplication. Presentations on Negations and Induction
For all c, there exists a,b in R s.t. c in C ---> c=a+b i
There exists a,b in R s.t. for all c, c in C ---> c=a+b i
There exists a,b in R s.t. there exists c, c in C ---> c=a+b i
For all x in R, there exists y in R s.t. xy=1.
There exists y in R s.t. for all x in R, xy=1.
There exists x in R s.t. for all y in R, xy=1.
For all n in N, 2^(2n) -1 is divisible by 3.
1^2 +3^2 +5^2 +...+(2n-1)^2 = n(2n-1)(2n+1)/3 for all n in N
2^2 +4^2 +5^2 +...+(2n)^2 = 2n(n+1)(2n+1)/3 for all n in N
1/(1*2) + 1/(2*3) +1/(3*4) =... = 1/(n*n+1) = n/(n+1)
For all n in N, 2 divides 3^n-1
For all n in N, 5 divides 8^n-3^n HW for Fri Work on Dr. Sarah's Maple Demo on 1-1 and Onto Functions, work on PS 5
• Fri March 10Sophie Germain Maple Demo, HW for spring break Work on PS 5 revs, Skim p. 1-51 in the book,
read p. 51-56 and do p. 56 #1,
Find the flaw in the proof that all horses are the same color. (Hint, in the induction step, trace through that part of the proof for specific small numbers k).
review demo on Sophie Germain's work on FLT,
and prove or disprove each of the following:
For all x not 0 in R, there exists y in R s.t. xy=1.
There exists y in R s.t. for all x not 0 in R, xy=1.
There exists x not 0 in R s.t. for all y in R, xy=1.
Cayley, Cauchy, Browne, and Jordan, prepare presentations.
• Mon March 20Review group def, discuss horses are the same color induction proof flaw, Sophie Germain demo, collect p. 56 #1 and
For all x not 0 in R, there exists y in R s.t. xy=1.
There exists y in R s.t. for all x not 0 in R, xy=1.
There exists x not 0 in R s.t. for all y in R, xy=1.
Define the order of a group, abelian group. Prove that GL(2,R) is an abelian group under matrix addition, and that it is not abelian and not a group under matrix multiplication. HW for WedShow that GL(3,R) is an abelian group under matrix addition
• Wed March 22Hand back collected work and solutions from Monday's collected work, go over the fact that the order in a proof must match the statement exactly for it to be correct (although scratchwork might be very different), presentation on Browne, reason why GL(2,R) is not a group under matrix multiplication, and then formal proof of this. HW for FriProve that GL(2,R) is not abelian. See handout on winning a million dollars if you solve Goldbach's conjecture!
• Fri March 24Go over Wed homework, intro to WebCT, online quiz, bulletin board, cyclic groups, direct product of groups and Maple demo on application of direct product of cyclic groups to Data Security. HW for Mon Review Maple demo on application of direct product of groups and post answers to WebCT bulletin board - Forum - Maple Demos, Start working on PS 6.
• Mon, March 27 Review defs of group, subgroup, cyclic group, direct product of groups, Z2, Maple demo from Fri, proof that GL(2,R) is not abelian. HWStudy for Friday's test (see main web page for material to study), work on PS 6.
• Wed, March 29Presentations on Kronecker and Sylow
• Fri, March 31 Test 2 on WebCT (see main web page for study suggestions) HW for Monday Redo Webct Test for Mon at 11pm, PS6 draft 1 due Tues at 5 I will be answering bulletin board questions Saturday and Sunday
• Monday, April 3Presentation on Gauss, Review Sylow's First Theorem, The Fundamental Theorem of Finite Abelian Groups, and The Fundamental Theorem of Algebra
• Wed, April 5Start Rings - Definition of a Ring, examples - real numbers, integers, and real numbers \{0}. See WebCT for review. HW for Fri Continue working on next draft of PS 6 due Monday, study for WebCT quiz on Fri (bios Browne, Kronecker, Gauss, and Sylow, and the definition of a group).
• Fri, April 7Quiz on groups, Rings Continued-work on examples in groups (see WebCT bulletin board Examples).
• Mon, April 10Review of a ring, review of subgroups and direct sum of groups via examples from Z6 of Sylow Theorem, and Kroneckers Fundamental Theorem of Finite Abelian groups, Definition of a subring, {0,3}=Z2 under + and * mod 6 is a subring of Z6, definition of direct sum of rings as a ring, and Z2directsumZ3 is a ring, Break up into groups - each group examines one of the following statements for rings:
a,b,c in R, a not 0, ab=ac implies b=c
a,b, in R, ab=0 implies a=0 or b=0
a in R implies, a not 0, a^2 = a implies a=0 or a=1.
1) Write out the hidden quantifiers in the statement
2) Write out the negation of the statement
3) Prove the negation.
4) Prepare presentation for Wednesday
• Wed, April 12 Presentations from Mon, Presentation on Hamilton, definition of field and integral domain HW for Fri Review my comments on your WebCT postings on ring examples, study for Friday quiz (see main web page for study suggestions).
• Fri, April 14 WebCT quiz 3 on groups and rings, work in groups on your example from last Friday - Is your example an integral domain? a field? Prove your claims. Does your example contain a non-trivial subring that is an integral domain or a field? Explain in detail. Post your classwork on the WebCT bulletin board - Forum - examples. HW for Mon Complete proofs from longer matching problem on WebCT quiz 3, review WebCT postings in Forum Examples, work on PS 6 revisions due next Wed (you can post questions in the Forum Problem Set 6, which I will try to answer at least once a day), and retake WebCT quiz 3 till you score a perfect score.
• Mon April 17Hints on PS6 about tetrahedron, then fields and integral domains continued
• Wed April 19Presentations on Galois and Noether, hand out problem set 7 on rings, fields and integral domains
• Fri April 21Presentations on Dedekind, WebCT quiz on groups, rings, fields and integral domains, begin Maple Demo on Quaternions
• Wed April 26Symmetries of the Tetrahedron presentations with your models
• Fri April 28 WebCT quiz on groups, rings, fields and integral domains, finish Maple Demo on Quaternions, post answers to questions
-Write down similar Maple commands to show that the other 7 commands hold: j^2=-1, k^2=-1, ji=-k , jk=i, kj=-i, ki=j, ik=-j.
-Finish this proof that H is not abelian
on WebCT Bulletin Board -Forum- Maple Demos
• Mon May 1Presentation on Lie, review Maple demo on quaternions, discuss relationship of quaternions to rotations of the space shuttle.
• Wed May 3 Review Game
• Tues May 9 3-5 Final Exam