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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 SPRING 2026 WITH NOTES AND TEST SOLUTIONS
Use this to refer to what we did last time. We will follow the same schedule and topics.
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.



Started Aug 11, 2026.


TEST DATES:
TEST 2 OCT 21
TEST 3 NOV 18
FINAL DEC 8 2-4PM

TEST 3 Wed Nov 18, IN CLASS. REVIEW MONDAY Nov 16 IN CLASS.
WILL COVER ALL MATERIAL DONE AFTER TEST 2.
PLEASE STUDY THE CLASS NOTES, TRY THE PROBLEMS INSIDE THE NOTES AND PRACTICE PROBLEMS POSTED HERE (SEE BELOW),AND CLASSWORK, ALL AFTER TEST 2.

Try the problems first and then look at solutions:
Problems 1, 2 of the following tests: Fall 25 Test III Version 1 PROBLEMS; . Fall 25 Test III Version 1 Solutions. ;
Test III Version 2 PROBLEMS. ; Test III Version 2 Solutions
Problems 5, 6, 7 of the following test: Spring 26 Test III Version 1 PROBLEMS; . Spring 26 Test III Version 1 solution.
The last problem of these midterms from fall 25: Midterm Version 2,Midterm Version 1 and these two quizzes from fall 25: Quiz 4 version 1, Quiz 4 version 2.
The practice problems on sequences and limits to be posted on friday. We will show how to prove that a sequence goes to zero or infinity using epsilon notation, and also how to use squeeze theorem.
For example, show that 1/2n, sin(n)/n go to zero as n goes to infinity.
You can see what we did in spring 26 by going to spring 26 update page.

11/13/2026 Friday
Today we started talking about sequences and limits.
A problem on squeeze theorem.
Practice problems: (Chapter titled Convergence) 16.4 (practice problem as well as exercises). Some of them can be done using definition of limit, some using squeeze theorem.
Notes on sequences and limits.

Monday, 11-9-26
Today we finished talking about strong induction.
Please read these notes on strong induction and try some of the problems.
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).

EXTRA READING:
We did some examples of the postage stamp problem aka Frobenius coin problem aka Chicken Mcnugget problem .
FACT (Challenging to prove): 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.
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. Can you tell where he uses strong induction? Actually, he avoids it. How?
Notes from my Discrete Structures class on Egyptian Fractions.


Friday, 11-6-26
Today we finished proof by induction and started strong induction.
We watched a video on Fibonacci numbers and the golden ratio and computed the golden ratio.
We started proof by strong induction by showing that the n-th Fibonacci number is always less than 2n-2.
Please read these notes on strong induction and try some of the problems.
Please also read try some of the problems in these Proof by Induction Notes
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).


Monday, 11-2-26
Today we did some problems involving proof by induction.
We proved that 7n-1 is always divisible by 6 and 52n-1 is always divisible by 8.
We also looked at the proof of the sum of arithmetic and geometric sequences.
Please read these notes and try some of the problems in it.
Proof by Induction Notes
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).

Some slightly challenging problems:
Show that 3n + 7n - 2 is always divisible by 8 for n greater than or equal to 1.
Show that if you draw n lines none of whom are parallel and no three of whom intersect at a point, then it divides the plane into (n2 + n + 2)/2 disjoint regions.

Friday, 10-30-26
Today we continued talking about proof by induction.
We proved Bernoulli's inequality.
We saw how to prove that countable union of countable sets is countable, using induction.
Please read these notes and try some of the problems in it.
Proof by Induction Notes
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).

Wednesday, 10-28-26
Today we continued talking about proof by induction.
We saw how to prove the formulae for adding natural numbers and whole numbers, using induction.
Please read these notes and try some of the problems in it.
Proof by Induction Notes
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).

Tuesday, 10-27-26
Yesterday we continued talking about proof by induction.
Please read these notes and try some of the problems in it.
Proof by Induction Notes

Friday, 10-23-26
Today we went over test 2 and started talking about induction.
We saw how the proof of "subset of a countable subset is countable" uses the well-ordering theorem AND induction, albeit induction process is implicit in it.

Wednesday, 10-21-26
Please read solutions for test. We will start new topic friday.
Test II Problems
Test II Solutions

TEST 2 on Oct 21 wednesday in class. REVIEW MONDAY OCT 19 IN CLASS.
Please sign up for oral exam 2 through link on canvas announcements.

PLEASE STUDY THE CLASS NOTES AND PRACTICE PROBLEMS POSTED HERE (SEE BELOW), CLASSWORK, AND THE FOLLOWING:
Test II will be similar in format to the spring semester Test II but with extra topics:
Try the problems first and then look at solutions:
Problems 4, 5, 6 of test 2 from spring 26;
Problems 3, 4 of test 3 from spring 26
Fall 25 Test I Version 2 PROBLEMS;
Fall 25 Test I Version 2 Solutions

Sample problems:
(NOTE: From text book 3rd ed ; some problems are labelled as practice problems and others as exercises.
The ones below are EXERCISES).

Exercises in functions: Chapter 2: 6.5 to 6.14 in section 6.
Exercises on cardinality :
Chapter 2, section 8, 8.1, 8.2, 8.3, 8.6 to 8.12.

Monday, 10-19-26
Today we reviewed for the test: equivalence classes and cardinality.
Please read these notes from class.

Friday, 10-16-26
Today we finished talking about countable and uncountable sets.
We went over the proof that countable union of countable sets is countable and used it to prove that algebraic numbers form a countable set.
Please read these notes from class.
Please also read these notes on relations and functions.
(THE NOTES HAVE BEEN UPDATED> PLEASE RELOAD THEM TO GET REVISED VERSION).
Please try exercises (NOT practice problems) on cardinality and countability:
Chapter 2, section 8, 8.1 to 8.12.

Wednesday, 10-14-26
PLEASE SIGN UP FOR ORAL EXAM 2 IF YOU HAVEN'T. SHEET IS ON CANVAS.
Today we talked about countable and uncountable sets.
We went over the proof that rational numbers are countable, real numbers and irrationals are uncountable, union of two countable sets is countable.
Please read these notes from class.
Please also read these notes on relations and functions.
(THE NOTES HAVE BEEN UPDATED> PLEASE RELOAD THEM TO GET REVISED VERSION).
Please try exercises (NOT practice problems) on cardinality and countability:
Chapter 2, section 8, 8.1 to 8.12.

Monday, 10-12-26
PLEASE SIGN UP FOR ORAL EXAM 2 IF YOU HAVEN'T. SHEET IS ON CANVAS.
On friday we spent some time discussing equivalence classes.
We went over the proof that equivalence relations on a set partition that set into disjoint equivalence classes and vice versa.
(This is theorem 6.15 in textbook).
Please read these notes from class.
Please also read these notes on relations and functions
and do some of the exercises on equivalence classes in it (especially examples 3 and 4).
(THE NOTES HAVE BEEN UPDATED> PLEASE RELOAD THEM TO GET REVISED VERSION).
Please try exercises (NOT practice problems) 6.7 to 6.14 on equivalence classes and partitions.

Wednesday, 10-7-26
Today we completed proof of the countability of integers, and proved that if a set can be mapped injectively into natural numbers or if the naturals map surjectively to a set, then such a set is countable.
The proof is found in these notes.
We also showed that the cartesian product of natural numbers with itself, namely the set of all pairs (m,n) with m,n being natural numbers (positive integers) is countable.
Please also read these notes on relations and functions.
Exercise: Use the idea of the proof that that the cartesian product of natural numbers with itself is countable to prove that cartesian product of ANY two countable sets is countable.

Monday, 10-5-26
Today we started discussing cardinality of infinite sets.
We showed that the even numbers, the perfect squares, and in general any subset of a countable set such as the natural numbers is countable.
We also showed the integers are in bijection with natural numbers using the map f(n) = 2n-1 if n is positive integer and f(n) = -2n+2 if n is negative integer or zero.
Show that it is a bijection:
It is 1-1 because if two positive integers go to same odd number, then 2m-1 = 2n-1 and so m = n and similarly for negative integers. (A positive integer cannot go to an even number and vice versa).
Similarly show that it is onto: If 2n-1 is odd, then it comes from n. What is the preimage of any even natural number?
Exercise: Show that the set of even integers is countable.

Friday, 10-2-26
Today we finished discussing equivalence classes and orbits.
Here are notes thanks to Cornell University on how Julia Sets (in the theory of fractals) are constructed using orbits.
Please also read these notes on relations and functions
and do some of the exercises on equivalence classes in it (especially examples 3 and 4).
(THE NOTES HAVE BEEN UPDATED> PLEASE RELOAD THEM TO GET REVISED VERSION).
Exercise: Show that equivalence relations partition the set and conversely, partition of a set gives an equivalence relation by "xRy iff x, y belong to same subset" (This is theorem 6.15 in textbook).
Please try exercises (NOT practice problems) 6.7 to 6.14 on equivalence classes and partitions.

Wednesday, 9-30-26
Today we discussed equivalence classes and orbits.
Here are notes thanks to Cornell University on how Julia Sets (in the theory of fractals) are constructed using orbits.
Please also read these notes on relations and functions
and do some of the exercises on equivalence classes in it (especially examples 3 and 4).
(THE NOTES HAVE BEEN UPDATED> PLEASE RELOAD THEM TO GET REVISED VERSION).
Exercise: Show that the relation "mRn iff m has the same number of divisors as n" is an equivalence relation on the set of natural numbers.
Find the equivalence classes of the set of natural numbers under this relation.
What is the equivalence class of 2 under this relation? Note that 2 has 2 divisors, namely 1 and itself.
What is the equivalence class of 4?

Monday, 9-28-26
Today we discussed equivalence relations with examples from exercise 6.5 in textbook.
What we did is in these notes.
Please also read these notes on relations and functions.
Please try exercises 6.5 and 6.6 from chapter 2 section 6.

Friday, 9-25-26
Today we continued talking about relations and functions, in particular equivalence relations with example from modular arithmetic.
Please read these notes on relations and functions.
Please try exercises 6.5 and 6.6 from chapter 2 section 6.

Wednesday, 9-23-26
Please read solutions for test. We will start new topic wednesday.
Test I Problems
Test I Solutions

TEST 1 on SEP 23 wednesday in class. REVIEW MONDAY SEP 21 IN CLASS.
Please sign up for oral exam 1 through link on canvas announcements.

It will be similar in format to the spring semester Test I but with extra topics:
Try the problems first and then look at solutions:
Problems 1,2,3 of test 2 from spring 26;
All Problems of test 1 from spring 26
Fall 25 Test I Version 1 PROBLEMS;
Fall 25 Test I Version 1 Solutions

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.

Sample problems:
(NOTE: From text book 3rd ed ; some problems are labelled as practice problems and others as exercises.
The ones below are EXERCISES).

Chapter 2 (sets) :5.3, 5.11, 5.15.
Exercises in functions: Chapter 2: 7.1, 7.2, 7.3, 7.9, 7.13, 7.16, 7.17.
EXERCISES FROM CHAPTER 1 :
1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12.

friday, 9-18-26
Today we continued talking about relations and functions, including what is 1-1 and onto and what is composition.
We saw examples of 1-1 functions that are not onto.
Please read these notes on relations and functions.
Exercises to try:
1. If f and g are surjective (onto), then f composed with g is surjective.
2. A function f has a well-defined inverse function (defined on its range) iff it is 1-1.
3. If A and B have the same (finite) number of elements, then any 1-1 function from A to B is onto and vice versa.
Chapter 1: Practice problem 1.6(a),
Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12, 5.3, 5.11, 5.15, 7.1, 7.2, 7.3, 7.9, 7.13, 7.16, 7.17 .

Wednesday, 9-16-26
Today we started talking about relations and functions, including what is 1-1 and onto and what is composition.
We saw how to prove that composition of two 1-1 functions is also 1-1.
Please read these notes on relations and functions.
Exercises to try:
1. If f and g are surjective (onto), then f composed with g is surjective.
2. A function f has a well-defined inverse function (defined on its range) iff it is 1-1.
Chapter 1: Practice problem 1.6(a),
Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12, 5.3, 5.11, 5.15, 7.1, 7.2, 7.3, 7.13, 7.16, 7.17 .

Monday, 9-14-26
Today we proved a problem by cases and reviewed some facts about sets.
Please read these notes on logical statements and proofs.
Also read this proof of DeMorgan's law for sets using same procedure.
Please study these old notes on set theory and logic.
Exercises to try:
1. Prove by cases: 2m+5n2 = 20 has no solution in positive integers.
2. Prove by contrapositive: If A is contained in B then complement of B is contained in the complement of A.
3. Show, by giving a proof by contradiction, that if 40 coins are distributed among nine bags so that each bag contains at least 1 coin, at least two bags contain the same number of coins.
4. Prove by cases: max(x,y) = [x+y + |x-y|]/2.
(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12, 5.3, 5.11, 5.15.

Saturday, 9-12-26
Today we proved a problem by cases plus direct proof, and another on sets also using direct proof.
Please read these notes on logical statements and proofs.
Also read this proof of DeMorgan's law for sets using same procedure.
Please study these old notes on set theory and logic.
Exercises to try:
Prove or give counterexample:
1. n2+n+1 is always odd.
2. If A is contained in B then their intersection equals A (we did part of this in class).
3. The set A-B is equal to the intersection of A and the complement of B.

(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12.

Wednesday, 9-9-26
Today we discussed a problem on nested quantifiers and direct proofs.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).

Exercises to try:
(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7, 4.3 to 4.12.

Friday, 9-4-26
Today we discussed quantifiers in statements with many variables.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).

Exercises to try:
Challenge problem: Show that if x > 0 then x^3 + (1/x^3) is always bigger than 2 but no other number (in other words, global minimum is 2). What about if you replace 3 by 4 or 5 or 6 or....?
(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7.

Thursday, 9-3-26
Yesterday we discussed negative of conditional statements, and quantifiers.
We did proof by counterexample and contrapositive as well.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included) on arguments and these notes on quantifiers.

Exercises to try:
1. Use contrapositive argument to show that x > 1, then 1/x < 1.
2. True or false: For all real numbers, x + (1/x) > 2.
(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7.

Monday, 8-31-26
Today we discussed conditional statements, converse, and contrapositive.
We showed equivalence of an if statement and its contrapositive, and NOT( p --> q) and p AND NOT(q), using truth tables.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).
Exercises to try, using same idea:
1. Prove equivalence of p --> q and NOT(p) OR q using truth table.
2. Write the negative, converse, and contrapositive of the following:
If the earth keeps warming the sea levels will rise.
Also try following practice problems and exercises on the textbook, about logical statements:
(NOTE: In textbook, practice problems are different from exercises!)
Chapter 1: Practice problem 1.6(a), Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13, 3.1, 3.2, 3.6, 3.7.

Friday, 8-28-26
Today we discussed DeMorgan's law for sets and logic, and using them to write negatives of statements.
We showed equivalence of NOT(p AND q) and NOT(p) OR NOT(q) using truth tables.
Exercises to try, using same idea:
1. Prove equivalence of NOT(p AND q AND r) and NOT(p) OR NOT(q) OR NOT(r) using truth table.
2. Write the negative of the following:
Either climate is not changing or sun is getting hotter or we are all living in a bad dream.

Wednesday, 8-26-26
Today we discussed a few things in mathematics that were very surprising, mysterious and even unsettling when first discovered.
We already talked about the possibility of non-Euclidean geometry, where angles in a triangle add up to more than 180.
Today we talked about the irrationality of square root of 2 and uncountability of the real numbers.
Exercises to try, using same idea:
1. Prove that the square root of 3 is irrational.
2. Prove that cube root of 2 is irrational.

(Thanks to Veritasium) Video about axiom of choice, starting with proof of uncountability of real numbers.

Monday, 8-24-26
Today we concluded proving, using the congruence rules for triangles, some basic geometry facts.
In particular we saw how to prove that diagonals in a rectangle are equal, and angles opposite the equal sides of an isosceles triangle are equal.
We concluded with a geometric proof of Pythagoras' theorem (see below for more on Pythagoras' theorem).
Exercises to try, using congruence of triangles:
1. Prove that the pairs of opposite angles formed at the intersection of two lines are equal.
2. Prove that in any parallelogram the diagonals biect each other.
There are more geometry exercises in these Notes from first class of 2019.


Sunday, 8-23-26
Friday we discussed Euclid's axioms, the congruence rules for triangles, and then saw how to prove some basic geometry facts using those rules.
In particular we saw how to prove that alternate angles formed by a line intersecting two parallel lines are equal, and then proved that angles in any triangle add up to 180 degrees.
We concluded with the formula for area of a parallelogram.
Please also read these Notes from first class of 2019.


Wednesday, 8-19-26
Today we discussed a bit more the emerging field of AI generated math.
Following a clip from Tao's talk (see below) we briefly discussed logic, set theory and axioms.
We also talked about proofs in general, and just to get started and get a flavor for proof we found the formula for sum of the first n terms of an arithmetic sequence based on gauss' idea.

Saturday, 8-15-26
In our first class we will introduce ourselves, talk about how the course will be structured and provide important information about things you need to pay attention to, throughout the semester.
In remaining time we will discuss the emerging field of AI generated math.

Some questions for you:
What does AI mean?
How does it work?
What are some of the things it can do?
What does it mean for society?

Try to listen to these two talks by Terry Tao, one of the best mathematicians around.
Talk at the International Conference of Mathematicians that just concluded in Philadelphia.
Terry Tao's talk on machine learning and mathematical proofs.


Tuesday, 8-11-26.
You can see what we did in the final exams and in our first class of last semester.
Final exam problems.
Final exam solutions.

Notes from first class of 2019.

We saw how to prove Pythagorean theorem using geometry.
Outline of the proof given by two high school students in New Orleans using trigonometry.

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?