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Update page--Discrete Structures-Fall 2025 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

Text: Discrete Mathematics by Richard Johnsonbaugh, 8th edition, Prentice Hall

Syllabus with schedule of classes and exams, grading scheme (SUBJECT TO CHANGE)

Notes and exams from fall 2025.




Started 8-11-2026


TEST DATES:
Midterm 1: Wednesday, September 30
Midterm 2: Wednesday, November 4
FINAL EXAM DEC 4 FRI, 3.30-5.30PM

Friday, 11-23-26
Please read solutions for quiz. We will start new topic monday.
Quiz 8 Problems
Quiz 8 Solutions

11/9/2026 Monday
QUIZ 8 on FRIDAY 11/13.
Please read notes and do problems from 6.2 and 6.3 posted below, today and last friday.
Today we talked about generalized permutations and combinations.
Notes and Classwork from today on generalized permutations and combinations.
Please also read the sections on permutations and combinations in the notes posted last friday, below.
Please try these problems on generalized permutations and combinations. (From 8th edition ; Make sure you are doing same kind of problems as in notes above).
6.3: 1, 4, 7, 10, 17, 18, 25, 39, 40, 45, 48.

Friday, 11-6-26
Today we talked about counting methods using permutations and combinations in situations such as finding at least 3 heads in 10 tosses or at most 3 heads in 10 tosses, etc.,
Please read these notes:
Notes on counting methods from fall 25.
Notes on counting methods from spring 2019 (IGNORE PROBABILITY).
Please try these practice problems:
6.2: 43, 46, 49, 52, 55, 61 to 68.

Wednesday, 11-4-26
Please read solutions for test. We will start new topic friday.
Test II Problems
Test II Solutions

TEST 2 (MIDTERM 2) on Wednesday, Nov 4.
CHAPTER 3 sections 3.1 to 3.4 and CHAPTER 6 section 6.1
STUDY CLASS NOTES, QUIZ SOLUTIONS, PRACTICE PROBLEMS POSTED HERE EVERYDAY
Please try first MIDTERM II Version 1 PROBLEMS (ALL EXCEPT 1 and 6) and then MIDTERM II Version 1 Solutions (all except 1 and 6).
ALSO: FALL 2017 Test II Solutions (only 1 and 3)
Problems on partitions: Find the equivalence relation and equivalence classes given by the following partitions:
Real numbers as union of the intervals [n, n+1) (that is, [1,2), [0,1), [-1,0), etc.,
Subsets of [1,2,3,...,9,10] divided into union of { } and the subsets {1}, [2}, ...,{10} and the subsets {1,2},{1,3},{2,3}....
Also study notes, quizzes, and practice problems below, posted after Midterm test 1.

Friday, 10-30-26
TEST 2 NEXT WED NOV 4. DETAILS ABOVE.
Today we talked about counting methods including permutations and combinations.
Please read these notes:
Notes on counting methods from fall 25.
Notes on counting methods from spring 2019 (IGNORE PROBABILITY).
Please try these practice problems:
6.2: 1, 4, 7, 10, 28, 33, 34, 39.

Wednesday, 10-28-26
TEST 2 NEXT WED NOV 4. DETAILS SOON.
Please read solutions for quiz. We will start new topic friday.
Quiz 7 Problems
Quiz 7 Solutions

Tuesday, 10-27-26
QUIZ 7 TOMORROW (WED 10/28). DETAILS BELOW.
Please study notes from last 3 classes (below), about partitions and equivalence classes, partial order and total order, and basics of permutations.
Yesterday we talked about counting methods including permutations.
We discussed the multiplication principle, and counting without and with repetition.
Notes on counting methods from fall 25.
Problems on partitions, partial order, and counting methods:
Please try exercises 1, 4, 7, 10, 13, 16 in section 6.1.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.
Also in section 3.3, problems 24, 27, 30, 33, 35, 37.
Also try in section 3.4, problems 1, 4, 7, 10, 11, 14, 17, 20, 27, 28, 32, 33, 34, 41, 43.

Friday, 10-23-26
Today we talked about antisymmetry and partial orders and also concluded talking about how equivalence classes partition the set into disjoint subsets and how a partition of a set automatically gives an equivalence relation.
The proof of this fact for any set.
The proof that Equivalence classes partition the set (the converse of the previous statement about partitions giving rise to equivalence relation) is in these notes.
We also talked about partial and total orders and antisymmetry.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.
Also in section 3.3, problems 24, 27, 30, 33, 35, 37.
Also try in section 3.4, problems 1, 4, 7, 10, 11, 14, 17, 20, 27, 28, 32, 33, 34, 41, 43.

Wednesday, 10-21-26
Today we discussed how equivalence classes partition the set into disjoint subsets.
We also talked about how a partition of a set automatically gives an equivalence relation.
For example, suppose we divide HU students based on their majors, and no one has a double major, then "x has the same major as y" gives an equivalence relation on the set of students of HU.
The equivalence classes are just students with the same major.
The proof of this fact for any set.
The proof that Equivalence classes partition the set (the converse of the previous statement about partitions giving rise to equivalence relation) is in these notes.
Please also read these notes these notes from last friday's class on equivalence classes.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.
Also try in section 3.4, problems 1, 4, 7, 10, 11, 14, 17, 20, 27, 28, 32, 33, 34.

Monday, 10-19-26
Please read solutions for quiz. We will start new topic wednesday.
Quiz 6 Problems
Quiz 6 Solutions

friday, 10-16-26
QUIZ 6 MONDAY 10/19, on equivalence relations and sequences.
Today we discussed equivalence classes.
Will put zoom recording of class on canvas.
Please read these notes these notes from class on equivalence classes.
Please also read the part on equivalence relations in these notes.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.
Also try in section 3.4, problems 1, 4, 7, 10, 11, 14, 17, 20, 27, 28, 32, 33, 34.

Wednesday, 10-14-26
QUIZ 6 MONDAY 10/19, on equivalence relations and sequences.
Today we discussed sequences.
Please read these notes these notes from class on sequences and strings.
We won't spend much time in class on strings but it is easy to read the material in the text.
Please also read the part on equivalence relations in these notes.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.
Also try in section 3.4, problems 1, 4, 7, 10, 11, 14.

Monday, 10-12-26
On friday we discussed sequences.
Please read these notes these notes from class on sequences and strings.
We won't spend much time in class on strings but it is easy to read the material in the text.
Please try exercises 45 to 64 and 88 to 100 in section 3.2.

Wednesday, 10-7-26
Please read solutions for quiz. We will start new topic friday.
Note that, in problem 1, the domain is non-zero real numbers. So we are able to divide by any number and cancel out the denominators.
If the domain were all real numbers, you would have to be careful with cancellation.
Quiz 5 Problems
Quiz 5 Solutions

Monday, 10-5-26
Quiz 5 will be wednesday Oct 7. It will cover Euclidean algorithm and basics of functions such as 1-1 and onto.
Today we discussed the definitions of functions, 1-1 (injective) and onto (surjective) functions, and inverse functions.
Please try section 3.1 exercises 17, 20, 23, 26, 30, 33, 35, 43, 45, 46.

Friday, 10-2-26
Quiz 5 will be next wednesday Oct 7.
Today we discussed the well ordering theorem and its applications.
We watched a little bit of video about Cantor's efforts to show that real numbers are well orderd (they are not!).
EXERCISE 1: Try to prove that, for any two positive integers a,b, GCD(a,b) = ax+ny for some integers x,y, USING well ordering theorem.
If you cannot get it after trying for a while, look at the proof:
The proof is in these notes.
EXERCISE 2: Find the GCD of 20 and 15 using Euclidean algorithm.
The Euclidean algorithm and the GCD are discussed in section 5.3 of textbook.
5.3 also has proof of Exercise 1 using the steps of Euclidean algorithm.

Wednesday, 9-30-26
Please read solutions for test. We will start new topic friday.
Test I Problems
Test I Solutions

TEST 1 (MIDTERM 1) on Wednesday, Sep 30.
ALL OF CHAPTERS 1 & 2 (BOTH 7TH & 8TH ED) EXCEPT 2.3 (resolution proofs)
STUDY CLASS NOTES, CLASSWORK NOTES, QUIZ SOLUTIONS, PRACTICE PROBLEMS POSTED HERE EVERYDAY
Please try first MIDTERM I Version 1 PROBLEMS and then MIDTERM I Version 1 Solutions.
ALSO: FALL 2017 Test I Solutions
Fall 04 Test I (all except 5)
Fall 06 Test I (all except 7)
Also study these class notes (with solutions included).

Friday, 9-25-26
Today we did some problems on proof by strong induction.
Please read these Notes on Egyptian Fractions (Problem 25, in 2.5).
The rest are in these Notes on Strong Induction.
Please study these old notes on proof by induction and try the exercises in it.
Also study these class notes (with solutions included).
Please read sections 4 of chapter 2 of the textbook (see at the top) on methods of proof:
(mostly the ones marked in blue ; the exercises are about proof by induction) :
2.5: 1, 3, 23.

Wednesday, 9-23-26
Today we did a problem on proof by induction and then started talking about strong induction.
Please read Notes on Strong Induction.
Please study these old notes on proof by induction and try the exercises in it.
Also study these class notes (with solutions included).
Please read sections 4 of chapter 2 of the textbook (see at the top) on methods of proof:
(mostly the ones marked in blue ; the exercises are about proof by induction) :
2.4: 1, 4, 7, 13, 16, 18, 19, 22, 26.

Monday, 9-21-26
Please read solutions for quiz. We will start new topic wednesday.
Quiz 4 Problems
Quiz 4 Solutions

Friday, 9-18-26
QUIZ 4 on Monday 9/21. It will cover proof by cases and proof by induction.
Please try problems from 2.2 and 2.4 mentioned this week, below.

Today continued discussing proof by induction.
Please study these old notes on proof by induction and try the exercises in it.
Also study these class notes (with solutions included).
Please read sections 4 of chapter 2 of the textbook (see at the top) on methods of proof:
(mostly the ones marked in blue ; the exercises are about proof by induction) :
2.4: 1, 4, 7, 13, 16.

Wednesday, 9-16-26
QUIZ 4 on Monday 9/21. More details soon.
Today started discussing proof by induction.
Please study these old notes on proof by induction and try the exercises in it.
Also study these class notes (with solutions included).
Please read sections 4 of chapter 2 of the textbook (see at the top) on methods of proof:
(mostly the ones marked in blue ; the exercises are about proof by induction) :
2.4: 1, 4, 7.

Monday, 9-14-26
QUIZ 4 on Monday 9/21. More details soon.
Today we saw some ways to prove mathematical statements using proof by cases.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).
Please read sections 2 of chapter 2 of the textbook (see at the top) on methods of proof:
(mostly the ones marked in blue ; the exercises are about proof by cases) :
2.2: 21, 22, 32, 36, 37, 39, 42.

Saturday, 9-12-26
Please read solutions for quiz. We will start new topic friday.
Quiz 3 Problems
Quiz 3 Solutions

Wednesday, 9-9-26
QUIZ 3 on Friday, covering 1.6, 2.1, 2.2
See practice problems below from today and last friday.
Today we saw some ways to prove mathematical statements involving numbers and sets: proof by contrapositive, direct proof.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).
Please read sections 1 and 2 of chapter 2 of the textbook (see at the top) on proofs:
(mostly the ones marked in blue ; the exercises are about methods of proof such as direct proof and proof by contrapositive) :
2.1: 7, 10, 14, 16, 21, 24, 33.
2.2: 1, 2, 3, 4, 5, 8, 9.

Friday, 9-4-26
Today we talked about nested quantifiers.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).
Please read section 6 of chapter 1 of the textbook (see at the top) on nested quantifiers:
(mostly the ones marked in blue ; the exercises are about nested quantifiers) :
1.6: 39, 40, 41, 42, 43, 48,60, 63.

Thursday, 9-3-26
Please read these notes on quantifiers.
(We did this in class on monday).

Please read solutions for quiz. We will start new topic friday.
Quiz 2 Problems
Quiz 2 Solutions

Monday, 8-31-26
QUIZ 2 on wednesday, 9/2.

Today we saw how to check validity of arguments, and quantifiers.
Please study these old notes on set theory and logic.
Also study these class notes (with solutions included).
Please read section 4 and 5 of chapter 1 of the textbook (see at the top) on arguments and quantifiers:
(mostly the ones marked in blue ; the exercises are about validity of arguments, and quantifiers) :
1.5: 39, 43, 49, 50, 55, 76.
1.4: 15, 18, 30.

Friday, 8-28-26
QUIZ 2 on wednesday, 9/2.

Today we discussed conditional statements, converse, and contrapositive.
Please study these old notes on set theory and logic.
Please read section 3 of chapter 1 of the textbook (see at the top) on basic notions of sets and logical statements do the following exercises:
(mostly the ones marked in blue ; the exercises are about conditional statements) :
1.3: 13, 39, 42, 73, 76.

Wednesday, 8-26-26
Please read solutions for quiz. We will start new topic friday.
Quiz 1 Problems
Quiz 1 Solutions

Monday, 8-24-26
QUIZ 1 on wednesday, 8/26.

Today we discussed DeMorgan's law for sets as well as logic.
We also talked about Venn diagrams, logical statements and did some problems.
Please study these old notes on set theory and logic.
Also read these Worked out examples from fall 25.
Please read section 1 and 2 of chapter 1 of the textbook (see at the top) on basic notions of sets and logical statements do the following exercises:
(mostly the ones marked in blue ; the exercises are about set operations such as union, intersection, etc.,) :
1.1: 1, 4, 7, 10, 13, 16, 17, 23, 35, 47, 61-65, 68, 93.
1.2: 16, 20, 23, 37, 56, 59.

Sunday, 8-23-26
QUIZ 1 on wednesday, 8/26. More details in class.

On friday we discussed briefly about paradoxes, comparing the famous paradoxes of Bertrand Russell and Schrodinger.
We then went over the basics of sets. Please study these old notes on set theory and logic.
Please read section 1 of the textbook (see at the top) on basic notions of sets and do the following exercises:
(mostly the ones marked in blue ; the exercises are about set operations such as union, intersection, etc.,) : 1.1: 1, 4, 7, 10, 13, 16, 17, 23, 35, 47, 61-65, 68, 93.

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 then talked briefly about paradoxes, starting with the famous paradox of Bertrand Russell.
On friday we will go over the basics of sets. Please study these old notes on set theory and logic.

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, 8-11-2026
Final exam solutions from last fall. Final exam Version 3 PROBLEMS.
Final exam Version A Solutions.
Final exam Version 3 SOLUTIONS.
Final Exam Version B Solutions.
What we did in first week of fall 2025:
We introduced logic and set theory. We talked about Russell's paradox, how Euclid's parallel lines postulate (that they never intersect) could not be proved and became an an axiom, and basic set theory.
Notes from class on set theory and logic .
We read section 1 of the textbook on basic notions of sets and did as many exercises as possible.

(Optional) Extra reading related to class:
Here is a How Russell's paradox means there is no set contining all sets
Note on Russell's paradox from 2017.
This note contains its definition, history, and an example using web search principles.

Terry Tao's talk on machine learning and mathematical proofs.

The Dunning-Kruger effect which is about people not knowing what they don't know.