1. Lecture 1 CS 1813 – Discrete Mathematics
Learning Goals
Lesson Plans
and
Logic
Rex Page
Professor of Computer Science
University of Oklahoma
EL 119 –
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. CS 1813 Discrete Mathematics Learning Goals
Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
CS 1813 Discrete Mathematics Learning Goals
Apply mathematical logic to prove properties of software
Predicate calculus and natural deduction
Boolean algebra and equational reasoning
Mathematical induction
Understand fundamental data structures
Sets
Trees
Functions and relations
Additional topics
Graphs
Counting
Algorithm Complexity
proofs galore!
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
Why Proofs?
100s of inputs
> 2100s of
possibilities
input signals
software
output signals
Key presses
Mouse gestures
Files
Databases …
computation
Images
Sounds
Software translates input signals to output signals
A program is a constructive proof of a translation
Responsible software developers are obligated specify what properties their software is guaranteed to have.
Proofs play a big role in this. Other methods of confirming software properties, especially testing, leave software riddled with errors.
Most systems have hundreds of input signals. Since each input signal must have at least two possible values, the number of different input states that must be checked to verify the operation of the software through testing is at least 2 to the power 100.
If you can generate, run, and examine one test case every minute, round the clock, you will not finish testing before you die.
Not even before the sun runs out of fuel. Not even if you do one test case every nanosecond.
Forget it. Testing can work only on very small pieces of software.
But what translation?
Proofs can confirm that software works correctly
Testing cannot confirm software correctness
Practice with proofs improves software thinking
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. CS 1813 Discrete Mathematics Textbook and Tools
Discrete Mathematics Using a Computer
Cordelia Hall and John O’Donnell
Springer-Verlag, January 2000
Tools provided with textbook
Download from course website for CS 1813
Hugs interpreter for Haskell
Download from course website
Haskell is a math notation (and a programming lang)
Reading assignments begin with Chapter 2
Read Chapter 1 (Haskell) as needed, for reference
Haskell coverage JIT, like other math notations
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. Formal Mathematical Notations
Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
Formal Mathematical Notations
Notations introduced as needed (JIT)
Logic ab, ab, ab, x.P(x), x.Q(x), 
Sets A  B, A  B, {x | xS, P(x)}, 
Sequences [x | x <- s, P(x)]
[4, 7, 2] ++ [3, 7] == [4,7,2,3,7]
Haskell
s(a: xs) = s[x | x <- xs, x < a]
++ [a] ++
s[x | x <- xs, x >= a]
The last one contains a sequence, but is not itself a sequence. It is an algebraic datatype using several constructors (Theorem, Imp, Not, etc.).
Structures Theorem [P, Q] (And P Q)
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
Coursework
Class Attendance REQUIRED
Reading assignments
See syllabus on course website
Study prior to class
Class Participation
Homework problem sets
Approximately weekly
Midterm Exam 1
Midterm Exam 2
Final Exam
10%
20%
Contribution to grade
40%
The Q/A Lab will be a lecture/workshop format.
The instructor will not come with a prepared agenda. Students ask questions, and the instructor attempts to answer them.
The instructor will bring a laptop computer for addressing problems with software. Students may also bring laptops.
The Q/A session will go on as long as necessary, or until 10pm, whichever comes first.
Q/A Lab Attendance NOT REQUIRED
Q/A Lab – Thursdays 8:00pm, CEC 439
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. Tiling with Dominos a mathematical proof – just for practice
Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
Tiling with Dominos a mathematical proof – just for practice
Dominos – size matches board
checkerboard with
two missing corners
Problem
cover board with dominos
no overlapping dominos
no dominos outside board
How many squares on board?
Adapted from Singh, Fermat’s Enigma, Walker & Co, 1997
So, how many dominos will it take?
One domino covers how many red squares?
Let’s do a little planning before we start laying dominos on the checkerboard.
How many dominos is it going to take to do the job?
When a domino is place on the board, how many red squares and how many black squares does it cover?
How many red squares are covered when 30 dominos have been placed? How many black squares?
How many red squares are left to be covered?
There are only 30 red squares, so they’re all covered when 30 dominos have been placed. That leaves no red square for the last domino to cover.
The last two squares are black. The last domino cannot cover them because no two black square are next to each other.
31 dominos cover how many red squares?
How many red squares are there?
Yikes! What’s wrong here?
TILT
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. How To Find a Million Dollars using logic
Lecture 1 - CS1813 Discrete Math, University of Oklahoma
12/10/2018
How To Find a Million Dollars using logic
Three Doors
Behind one is a million dollars
Behind another is a Palm Pilot
Behind the other is a melting Popsicle
Where’s the jackpot?
Why not A?
Why not B?
Must be C, eh?
Bonus question: Where’s the Palm Pilot?
Door C speaks the truth – the Palm Pilot is behind A
Door B lies – it has a Popsicle, afterall A
Palm
Here B
Popsicle
Behind C C
Palm
Behind A
Adapted from Smullyan, The Lady or the Tiger, Times Books, 1982
so palm here
if $$$ here
… popsicle here
so C sign correct
It can’t be door A because the statement on the jackpot door is true, and door A says the Palm Pilot is behind it.
So, the jackpot can’t be behind door A.
It can’t be door B because if door B had the jackpot, it’s sign would be true, and the Popsicle would be behind C.
But if the Popsicle were behind C, the statement on C would be false, but it can’t be because there’s nowhere else but door A for the Palm Pilot.
So the jackpot can’t be behind door B.
TILT
Signs on Doors
$$$ door: true statement
Popsicle door: false statement
If it was so, it might be;
and if it were so, it would be:
but as it isn’t, it ain’t. That’s logic.
- Tweedledee
in Through the Looking Glass
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. Tracing a Square and Its Diagonals
Square + Diagonals
Problem
Start at any corner
Trace some line to another corner
Then trace from that corner to another
Keep going until all six lines are traced
Don’t trace any line more than once (crossing OK, but not retracing)
Solution revealed in the next lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. CS 1813 Discrete Mathematics, Univ Oklahoma
Homework #1
Problem under “Assignments” tab in course website
It’s a hard problem
You don’t have much mathematical apparatus, yet, to attack it
Grade based more on thoughtfulness and well-expressed ideas than on solutions
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. CS 1813 Discrete Mathematics, Univ Oklahoma
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page