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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 2025 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.



4/1/26 Wednesday
Today we started Proof by Induction.
We saw why it was not enough to say, for example, the union of any finite set of countable sets is countable because union of two sets SUT is countable by theorem, so then (SUT)UV will be countable for 3 countable sets by using the theorem again, and so on.
The "and so on" needs to be made precise and proved, by showing how you go from any step to the next step.
Please read these Notes on Induction and try some of the problems there.

3/30/26 Monday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we finished talking about Cardinality, and proved some theorems on bijections and ended with the proof of irrationality of 2 by contradiction.
We also proved that m^3+3n^2 = 36 has no integer solutions usng proof by cases, asnd also that n^3+n is even for any integer n.
We proved that Cartesian product of two countable sets, or even a union of countable collection of countable sets, is countable. For this we made a map from natural numbers onto the positive rationals by connecting them to points in the first quadrant, and this combined with Theorem 8.10 (see Theorem 1 of the notes on theorem 8.10 below).
Proof of Theorem 8.10 (modified from book).
(Updated-please refresh!) Notes on relations from Fall 25.
Prove the following :
Prove that the cube root of 2 and square root of 3 are irrational.
Prove by cases that n^2+n is always even.

3/25/26 Friday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we proved some theorems on bijections and ended with the proof of countability of rationals.
Specifically we made a map from natural numbers onto the positive rationals by connecting them to points in the first quadrant, and this combined with Theorem 8.10 (see Theorem 1 of the notes on theorem 8.10 below) helped us prove that positive rationals are countable. Using this and the fact that union of countable sets is countable we showed that rational numbers are countable.
Proof of Theorem 8.10 (modified from book).
(Updated-please refresh!) Notes on relations from Fall 25.
Prove the following :
The Cartesian product of two countable sets is countable (use same kind of map as the one we used to show that positive rationals are countable).
The countable union of countable sets is countable (Proof similar, also in book).

3/25/26 Wednesday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we proved some theorems on bijections and cardinality and ended with the proof of uncountability of irrationals.
Proof of Theorem 8.10 (modified from book).
(Updated-please refresh!) Notes on relations from Fall 25.
Prove the following :
The set of points on hyperbola y = 1/x other than points with integer valued x-coordinates is uncountable.
The set of positive rational numbers whose denominators are either 3 or 5 is countable.
The problem above can be done either upon using countabiity of rational numbers or not using their countability. Try both.
Try the exercises 8.3 in chapter 2, "Sets and Functions" section titled "Cardinality" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

3/24/26 Tuesday.
Please work on wikipedia assignments as mentioned in Canvas.

Yesterday we did some exercises on bijections and cardinality.
Solution for classwork monday 3/23.
(Updated-please refresh!) Notes on relations from Fall 25.
Prove that the irrationals are uncountable.
Try the exercises 8.3 in chapter 2, "Sets and Functions" section titled "Cardinality" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

3/20/26 Friday
Today we discussed bijections between a countable set and any of its infinite subsets, for example natural numbers and even numbers.
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.
(Updated-please refresh!) Notes on relations from Fall 25.
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).

3/18/2026 wednesday
Please read Test II solutions.
TEST II PROBLEMS.
TEST II SOLUTIONS.

TEST 2 Wed Mar 18, IN CLASS. REVIEW MONDAY FEB 9 IN CLASS.
WILL COVER ALL MATERIAL DONE AFTER TEST 1.
It will be similar in format to the fall semester Tests:
Try the problems first and then look at solutions:
Problems 1, 2, 3, 7b of the following tests: Test II Version 1 PROBLEMS; . Test II Version 1 Solutions. ;
Test II Version 2 PROBLEMS. ; Test II Version 2 Solutions
SPRING 19 QUIZ 6, SPRING 19 QUIZ 4.

PLEASE STUDY THE CLASS NOTES, TRY THE PROBLEMS INSIDE THE NOTES AND PRACTICE PROBLEMS POSTED HERE (SEE BELOW),AND CLASSWORK, ALL AFTER TEST 1.

3/6 Friday
Today we went over a problem that is in (Updated-please refresh!) these Notes on relations from Fall 25.
Basically, we showed that the relation mRn iff 5 divides m-n is an equivalence relation and found all the equivalence classes and showed that the equivalence classes are disjoint and cover all integers.


3/4/26 Thursday.
TOMORROW (FRIDAY) CLASS WILL HAVE 10 POINTS EXTRA CREDIT FOR CLASSWORK.
This is to help people who might have missed classwork. It would also help in preparing for test 2.
Please work on wikipedia assignments as mentioned in Canvas.

Yesterday we did some exercises on equivalence classes and partitions.
Solution for classwork wednesday 3/4.
(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 6.5, 6.7, 6.8, 6.9, 7.1, 7.2, 7.3, 7.13, 7.16, and 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

3/2/26 Monday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we talked about equivalence classes and partitions.
Solution for classwork today.
(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 6.5, 6.7, 6.8, 6.9, 7.1, 7.2, 7.3, 7.13, 7.16, and 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

2/27/26 Friday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we talked about equivalence classes aka orbits.
Solution for classwork today.
Check out these articles on Six degrees of separation, Erdos Number, and Bacon number.

In addition to exercises below, try to find all the equivalence classes for the following relations:
Below, $X$ represents the set on which the relation $R$ is defined:
1. X = Students at Howard ; xRy iff x has same major as y. Assume there are no double majors.
2. N = natural numbers 1,2,3,... ; mRn iff 3 divides m - n.
3. Z = set of integers ; xRy iff x+y = 2k for some k in Z.

(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 6.5, 7.1, 7.2, 7.3, 7.13, 7.16, and 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

2/25/26 Wednesday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we talked about the relation between the length of the repeating blocks in decimal expansion of fractions and the remainders of the powers of 10.
We then talked about equivalence relations, in particular the relation given by "x and y leave the same remainder."
In addition to exercises below, try to check if the following are equivalence relations:
Below, $X$ represents the set on which the relation $R$ is defined:
Say which of the properties Reflexive, Symmetric and Transitive are satisfied by each:
1. X = People at a party ; xRy iff x is taller than y.
2. N = natural numbers 1,2,3,... ; mRn iff m divides n.
3. Z = set of integers ; xRy iff x+y = 2k for some k in Z.

(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 7.1, 7.2, 7.3, 7.13, 7.16, and 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

2/23/26 Monday.
Please work on wikipedia assignments as mentioned in Canvas.

Today we did some exercises on basic concepts of functions and talked about floor and ceiling functions and the modulus operator function (remainder function).
(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 7.1, 7.2, 7.3, 7.13, 7.16, and 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

2/20/26 Friday.
Today we talked about inverse functions, composition of functions and proved that inverse functions exist iff function is 1-1.
We looked at example of composition and found its domain.
(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 7.1, 7.2, 7.3, 7.13, 7.16, 7.17 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions and compositions of functions.

2/19/26 Thursday.
Please work on wikipedia assignments as mentioned in Canvas.

Yesterday we talked about relations, functions, 1-1 and onto.
(Updated-please refresh!) Notes on relations from Fall 25.
Try the exercises 7.1, 7.2, and 7.3 in chapter 2, "Sets and Functions" in Lay's book, 3rd ed.
In case your edition has different numbers, look for the true and false questions on basic function properties and finding range of functions.

2/16/26 Monday.
The heating system in ASB-B has been fixed. On wednesday we will meet in usual room.

2/13/2026 friday
Please read Test I solutions.
TEST I PROBLEMS.
TEST I SOLUTIONS.

TEST I ON FRI FEB 13, IN CLASS. REVIEW MONDAY FEB 9 IN CLASS.

It will be similar in format to the fall semester Test I:
Try the problems first and then look at solutions:
Test I Version 1 PROBLEMS; . Test I Version 1 Solutions. ; Test I Version 2 PROBLEMS. ; Test I Version 2 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, 5.15.
Exercises in proofs: Chapter 1: 3.1, 3.2, 3.6, 3.7, 3.9(b), 3.9(c), 4.13.
Logical statements: Chapter 1: Exercises 1.3, 1.7, 2.3, 2.5, 2.7, 2.13.

2/11/26 Wednesday 4;40pm
Please work on wikipedia assignments as mentioned in Canvas.

Today we talked about sets, including DeMorgan's law for sets.
This proof is for your reference ; it is not included for the test.
(Updated-please refresh!) Notes on sets from Fall 25.
Try the exercises titled "Some exercises in set theory" in page 12.

2/11/26 Wednesday, 10:25am.
Because our heating system in ASB-B is not working well, class will be on zoom today. Zoom link is in the announcements.
TEST 1 WILL BE IN CLASS ON FRIDAY. I will be in office until 4pm today if you need anything.
What we do in class today will not be on friday's test.
We will have a poll in class today.

2/9/26 Monday
Please work on wikipedia assignments as mentioned in Canvas.

Today we did a review for test 1. Whiteboard has been posted on canvas.

2/6/26 Friday
Please work on wikipedia assignments as mentioned in Canvas.

Today we talked about sets, including DeMorgan's law for sets.
This proof is for your reference ; it is not included for the test.
Notes on sets from Fall 25.
Try the exercises titled "Some exercises in set theory" in page 12.

2/4/26 Wednesday
Please work on wikipedia assignments as mentioned in Canvas.

We talked about negatives of logical statements, converse and contrapositive and quantifiers. We also did proof by contradiction and contrapositive.
Please try all the exercises in the notes below. Updated notes on Logic statements
Notes on logical statements and proofs from Fall 25.

2/3/26 Tuesday
Please work on wikipedia assignments as mentioned in Canvas.

We talked about negatives of logical statements, converse and contrapositive and quantifiers. We also proved a small lemma using contradiction.
Please try all the exercises in the notes below. Updated notes on Logic statements

1/30/26 Friday
Please work on wikipedia assignments as mentioned in Canvas.

We talked about negatives of logical statements, converse and contrapositive.
Today we discussed the equivalence of the conditional statement "A implies B" and "(NOT A) OR B."
We also saw how the negative of "A implies B" is "A AND (NOT B)" and not "A implies (NOT B)."
We proved equivalence of some of these using truth tables.
Please try all the exercises in the notes below. Updated notes on Logic statements

1/28/26 Wednesday
Please work on wikipedia assignments as mentioned in Canvas.

We talked about negatives of logical statements, converse and contrapositive.
Please try all the exercises in the notes below. Updated notes on Logic statements

1/23/26 Friday
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Today we discussed various ways to make misleading graphs or arguments.
Notes from fall 25 on logic
Some examples of strange correlations between random things
Discussion of misleading claims on vaccines and autism
My notes from another class about ways graphs can be misleading
We then looked at some statements to understand their logic.
Logic exercises -- try all of them!

1/21/26 Wednesday
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Today we discussed seven developments in math that at first caught people by surprise:
1. Square root of 2 (and other non-perfect powers) is irrational.
2. Euclid's fifth postulate couldn't be proved using the first four, leading to non-Euclidean geometry.
3. Inability to find well-ordering of real numbers (here is the story of how it affected Cantor and lead to Axiom of Choice).
4. Inability to prove that real numbers are the next level of infinite after countable sets (such as integers) and how it led to independence of Continuum Hypothesis.
5. Godel's Incompleteness theorem showing limits of logic.
6. Hilbert's tenth problem asking for general algorithm to determine existence of integer solutions to any polynomial leading to proof of Undecidability.
7. Russell's paradox in logic leading to Zermelo-Fraenkel's axioms.
We then looked at some statements to understand their logic.
Logic exercises -- try all of them!

1/16/26 Friday
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Today we discussed five developments in math that at first caught people by surprise:
1. Square root of 2 (and other non-perfect powers) is irrational.
2. Euclid's fifth postulate couldn't be proved using the first four, leading to non-Euclidean geometry.
3. Inability to find well-ordering of real numbers (here is the story of how it affected Cantor and lead to Axiom of Choice).
4. Inability to prove that real numbers are the next level of infinite after countable sets (such as integers) and how it led to independence of Continuum Hypothesis.
5. Godel's Incompleteness theorem showing limits of logic.
6. Hilbert's tenth problem asking for general algorithm to determine existence of integer solutions to any polynomial leading to proof of Undecidability.
We then looked at some statements to understand their logic.
Logic exercises -- try all of them!

1/15/26 Thursday
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Yesterday we discussed in detail the proof of the Pythagoras' theorem. Please read the notes below:
Notes from first class of 2019.
Geometric proof of Pythagoras' theorem is in page 10. Please also study the other geometry problems done in these notes. They could come up in test.
The full justification for Pythagoras theorem geometric proof given above, in notes from 2019 .
We then solved couple of problems involving the rhombus.
Outline of proof

1/12/26 Monday
Today we got to know each other and talked about the course.
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Also try to form of group with 2 other students. We will start working in groups on solving problems and writing proofs on wednesday.
We also talked about the proof of the Pythagoras' theorem. Please watch the videos and read the notes below:
Outline of the proof given by two high school students in New Orleans using trigonometry.

Notes from first class of 2019. Geometric proof of Pythagoras' theorem is in page 10.
The full justification for Pythagoras theorem geometric proof given above, in notes from 2019 .

1/9/2026 Friday
PLEASE REGISTER IN WIKIPEDIA ASSIGNMENT AND GET STARTED!
REGISTRATION LINK.

Final exam solutions from the fall. Happy holidays!
Final exam Version 1 PROBLEMS.
Final exam Version 1 Solutions.
Final exam Version 2 PROBLEMS.
Final exam Version 2 Solutions.

Notes from our first class last semester

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?

Started 1/9/2026