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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
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.