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Welcome!
Here you will see info about what we did in class, upcoming exams, solutions for past exams, etc.,
CHECK AT LEAST ONCE A WEEK!!
COURSE PAGE WITH SYLLABUS, GRADING SCHEME ETC
UPDATE PAGE FROM FALL 2019 WITH NOTES AND TEST SOLUTIONS
Use this to refer to what we did last time. The order of the material will be different this time.
I may also add a few and modify some things as we go along.
Textbook:
Analysis with Introduction to Proofs by Stephen Lay, 4th edition, Prentice Hall.
TUTORING SCHEDULE
11/5/2025 Wednesday
Today we talked about monotone sequences.
Notes on limits of sequences and series
Sample problems: (Chapter titled Monotone Sequences) Exercise 18.3 (NOT practice problem)
11/3/2025 Monday
Today we reviewed sequences and proved squeeze theorem.
Notes on limits of sequences and series
Sample problems: (Chapter titled Convergence) Exercise 16.4 (NOT practice problem)
11/2/2025 Sunday
On friday we started talking about limits and proofs involving limits.
Notes on limits of sequences and series
Practice problems: (Chapter titled Convergence) 16.4, 16.7, 16.10 (practice problems, not exercises)
10/29/2025 Wednesday
Today we finished proofs by strong induction.
Notes on Proof by induction from 2020 (Try some of the exercises in it!)
Practice problems:
Show using regular induction that, if you add the first n Fibonacci numbers, you get one less than the n+2-th Fibonacci number.
For example, 1+1+2+3 = 7, and 7 = 8-1.
For your information: The postage stamp problem is also known as the Coin problem or the Chicken McNugget problem.
Here is a Wikipedia article about it.
(Challenging to prove) FACT: If the positive integers x and y have no common factors, all numbers bigger than xy-x-y (but not xy-x-y itself!) can be expressed using x and y.
Slightly less challenging: Here is nice write-up by Nam Nguyen of Williams College on Egyptian fractions.
It contains the proof that the "greedy algorithm" for writing Egyptian fractions works.
He avoids using strong induction but can you prove it using strong induction? (How does he avoids it?)
10/27/25 Monday
Today we proved that 5^(2n) - 1 is always a multiple of 8 by induction and an estimate for Fibonacci numbers using strong induction.
Please read these notes on induction.
Please also read Notes on strong induction.
Please also read Notes from today.
Please try the problems in the notes above before reading their solution.
10/22/25 Wednesday
Today we proved the formula for cardinality of power sets and Bernoulli's inequality ((1+x)^n > or = 1+nx) by induction.
Please read these notes on induction.
Please also read Notes from today.
Please try 10.4, 10.6, 10.7 (geometric sequence sum formula), 10.9, 10.10, 10.13 (inequalities on n! and 2^n), 10.18 (binomial formula) from textbook (Lay, 3rd edition) and also the problems in the notes above.
10/20/25 Monday
Today we started proofs by induction.
Please read these notes on induction.
Please also read Notes from today.
Please try 10.4, 10.6, 10.7 (geometric sequence sum formula) from textbook (Lay, 3rd edition).
10/18/2025 Thursday
Pleae read test solutions.
Test II Version 1 PROBLEMS.
Test II Version 1 Solutions.
Test II Version 2 PROBLEMS.
Test II Version 2 Solutions.
TEST 2 ON FRIDAY OCT 17. REVIEW WED OCT 15.
PLEASE STUDY THE CLASS NOTES AND PRACTICE PROBLEMS POSTED HERE (SEE BELOW), CLASSWORK, AND THE FOLLOWING:
Spring 2019 Quiz 6 ;
Problems 4 and 5 from Spring-2019 final ;
1,2,3,4,7,8,9,10 from 2018 review problems
Prove by contradiction: Cube root of 2 is irrational.
10/15/25 Wednesday
Today we did some problems to prepare for test 2.
Please read Notes from today.
10/10/25 Friday
Today we did some problems on proof by contradiction and proof by cases.
Prove by cases that m^3+2n^2= 36 has no solutions in positive integers.
Prove by cases that for all integers n, n^3+n is even.
10/9/25 Thursday
Yesterday we discussed some exercises on equivalence relations, and showed that union of countable sets and cartesian products are countable.
As application we proved that rational numbers are countable and irrationals are uncountable.
Proof is different from one in the text book.
See below for notes and practice problems.
Also try the following from book:
8.3.(Find bijection between sets like (0,1), [0,1), real numbers, etc., )
-- For map from (0,1) to real numbers use tan(x).
8.9: Show that set of algebraic numbers(solutions of polynomials with integer coefficients) is countable.
10/3/25 Friday
10/6/25 Monday
TEST 2 ON FRIDAY OCT 17. MORE DETAILS SOON.
Today we discussed some exercises on equivalence relations, and showed that union of countable sets is countable.
Proof is different from one in the text book.
See below for notes and practice problems.
10/3/25 Friday
Today we discussed bijections between a countable set and any of its infinite subsets.
Proof is also in the text book.
Check out this video on Veritasium about cardinality of infinite sets and the saga of Georg Cantor trying to prove the well ordering theorem.
Please read Notes on Relations (Updated: reload!).
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
These problems are from 3rd edition. Please make sure you are doing problems on same topic.
Chapter 2: Exercises 8.3, 8.6, 8.7 which are about comparing the cardinality of different sets and 6.5 all sections (some of these are mentioned towards the end in "Notes on Relations" above).
10/1/25 Wednesday
Today we discussed bijections between infinite sets.
Please read Notes on Relations (Updated: reload!).
Solutions for Classwork.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: Exercises 8.3, 8.6, 8.7, and 6.5 all sections (some of these are mentioned towards the end in "Notes on Relations" above).
9/29/25 Monday
Today we discussed equivalence relations.
Please read Notes on Relations (Updated: reload!).
Online Notes on Julia sets for your information ;
pdf of the same.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: Exercise 6.5 all sections (or at least the ones mentioned towards the end in "Notes on Relations" above).
9/24/25 Wednesday
Today we discussed equivalence relations.
Please read Notes on Relations (Updated: reload!).
Online Notes on Julia sets for your information ;
pdf of the same.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: Exercise 6.5 all sections.
9/22/25 Monday
Today we discussed relations and functions.
Please read Notes on Relations.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: 7.1, 7.2, 7.3, 7.8.
9/19/25 Friday
Today we discussed the test problems and finished talking about sets.
We proved directly that if A is a subset of B then AUB equals B.
Please try the exercises in Notes on Sets.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: 5.5, 5.7, 5.8, 5.9.
Prove using counterexample that (A+B)UC = A+(BUC) is not true using counterexample (+ means intersection).
9/17/2025 Wednesday
Pleae read test solutions.
Test I Version 1 PROBLEMS.
Test I Version 1 Solutions.
Test I Version 2 PROBLEMS.
Test I Version 2 Solutions.
TEST I ON WED SEP 17, REVIEW MON SEP 15.
PLEASE STUDY THE CLASS NOTES AND PRACTICE PROBLEMS POSTED HERE (SEE BELOW), CLASSWORK, AND THE FOLLOWING:
Problems 1, 2 from 2019 Quiz 1 ;
9-9-2019 problems ;
2019 Quiz 2;
Prove by contrapositive: If the square of a real number is irrational, the number itself is irrational.
2018 HW 2.
9/15/25 Friday
Tutoring schedule above.
Today we did a review of all the material covered so far. If you didn't come to class, please get somebody's notes and study them.
9/12/25 Friday
Today we continued talking about sets.
Please try the exercises in Notes on Sets.
Please try the following: (NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 2: 5.3, 5.15.
Prove using counterexample that (A+B)UC = A+(BUC) is not true using counterexample (+ means intersection).
9/10/25 Wednesday
Tutoring schedule above.
Today we proved that square root of 2 is irrational and started talking about sets.
Notes from 2019 on methods of proof.
Please try the exercises in Notes on Sets.
9/8/25 Monday
Today we did some exercises in proofs.
Notes on logical relations and proofs (Updated, reload!).
Please try the following practice problems and exercises on the textbook, about proofs:
(NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 1: Exercises 3.9(b), 3.9(c), 4.13.
Prove by contradiction: Square root of 2 is irrational.
You may assume every integer can be broken down in a unique way using its prime factors.
9/5/25 Friday
Today we talked about proof by contradiction.
Notes on logical relations and proofs.
Please try the following practice problems and exercises on the textbook, about proofs:
(NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 1: Exercises 4.11, 4.13.
Prove by contradiction: If x^2 > 1, then either x > 1 or x < -1.
9/3/2025 Wednesday
Today we talked about proof by counterexample and contrapositive and Fermat's theorem on sums of squares.
Notes on logic (updated, reload!).
Please try the following practice problems and exercises on the textbook, about proofs:
(NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 1: Exercises 3.1, 3.2, 3.6, 3.7, 3.9.
Optional problem: Show that 3, 5, and 7 are the only prime triples, that is, 3 consecutuve numbers that are primes.
8/31/2025
On friday we talked about converse and contrapositive and Fermat's theorem on sums of squares.
Notes on logic (updated, reload!).
Please try the following practice problems and exercises on the textbook, about logical statements:
(NOTE: In the text book, some problems are labelled as practice problems and others as exercises).
Chapter 1: Exercises 3.1, 3.2, 3.6, 3.7.
Optional problem: Show that 3, 5, and 7 are the only prime triples, that is, 3 consecutuve numbers that are primes.
8/28/2025
Yesterday we talked about converse and contrapositive.
Notes from today on logic (updated, reload!).
Please try the following practice problems and exercises on the textbook, about logical statements:
Chapter 1: Exercises 3.1, 3.2, 3.6, 3.7.
8/25/2025
Today we talked about DeMorgan's rules of logic and conditional statements.
Notes from today on logic (updated, reload!).
Please try the following practice problems and exercises on the textbook, about logical statements:
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13.
8/22/2025
Today we talked about the disk rolling problem (see 8/18 below), the proof of the Pythagorean theorem, and basic logic operations.
Notes from Introductory class (updated, reload!).
Notes from today on logic.
Detailed proof of Pythagorean theorem.
8/20/2025
Today we talked about what proof means and different kinds of Proofs.
Notes from class today.
We saw how to prove Pythagorean theorem using geometry.
Outline of the proof given by two high school students in New Orleans using trigonometry.
8/18/2025
Today we introduced each other and talked about class in general.
Fun question to think about, before next class:
We saw that the moon rotates exactly one time around its own axis as it does one full rotation around the earth.
Note that, as it goes around the earth, it is facing the same way.
Suppose it also rotates around its own axis exactly once at the same time.
So every part of the moon will be facing the earth at some point during a single rotation of the moon around the earth.
This time how many rotations around its axis in one rotation around the earth?
Now imagine two circular disks (maybe coins) touching each other on a flat surface.
If one os smaller and of radius 1 unit, and the other is of radius 2 units, how many
times does the smaller disk rotate about its own axis as you roll it around the larger disk, always keeping them in contact?
Started 8/13/2025
Notes from first class of Fall 19
Please read and try problems on last page.