Slide 1
Slide 1 text
cs2102: Discrete Mathematics
Class 1: What makes computers go?
David Evans
University of Virginia
Slide 2
Slide 2 text
What makes it go?
Slide 3
Slide 3 text
For example, there was a book
that started out with four
pictures: first there was a
windup toy; then there was an
automobile; then there was a
boy riding a bicycle; then there
was something else. And
underneath each picture it said,
“What makes it go?”
From Richard Feynman, Surely You’re Joking, Mr. Feynman!
Slide 4
Slide 4 text
I thought, “I know what it is: They’re going to talk
about mechanics, how the springs work inside the
toy; …”
It was the kind of thing my father would have talked
about: “What makes it go? Everything goes because
the sun is shining.” …
“No, the toy goes because the spring is wound up,” I
would say. “How did the spring get wound up?" …
“I wound it up."
“And how did you get moving?"
“From eating."
“And food grows only because the sun is shining. So
it’s because the sun is shining that all these things are
moving.”…
From Richard Feynman, Surely You’re Joking, Mr. Feynman!
Slide 5
Slide 5 text
What makes it go?
Slide 6
Slide 6 text
Sun shines
Plankton thrives
Fossilizes
Slide 7
Slide 7 text
Electricity Transistors, Memory
Slide 8
Slide 8 text
What makes it go?
Slide 9
Slide 9 text
Transistors, Memory
Programs
Mathematics
Algorithms
Slide 10
Slide 10 text
Focus of cs2102
This course is about the underlying mathematics that makes computers work.
cs2102
cs3102 cs4102
Theory of Computation:
What Can and Cannot Be Computed
Algorithms:
Designing and Analyzing Information Processes
Slide 11
Slide 11 text
Developing Mathematical Skills
Precise Tools for Thinking:
Definitions
Proofs
Before Friday (questions on Survey):
“Habits of highly mathematical people”
Slide 12
Slide 12 text
Developing Mathematical Skills
Precise Tools for Thinking:
Definitions
Proofs
Mathematical objects used in Computing:
Slide 13
Slide 13 text
Developing Mathematical Skills
Precise Tools for Thinking:
Definitions
Proofs
Mathematical objects used in Computing:
Numbers, Sets, Graphs
Slide 14
Slide 14 text
Join the course slack group:
csmath.slack.com/signup
Some of you received an
invitation with a link if you
are enrolled/waiting for
class. Please accept the
invitation now (and then I’ll
invite the rest of the class).
Slide 15
Slide 15 text
Join the course slack group:
csmath.slack.com/signup
Post any questions/comments you have so far in #general
Answer in #inclass: why does computer
science mostly use discrete math?
Slide 16
Slide 16 text
Why Discrete Math?
Slide 17
Slide 17 text
Continuous Discrete
Slide 18
Slide 18 text
Continuous Discrete
There is something between
any two numbers
There is nothing between
adjacent numbers
Real numbers Integers
Slide 19
Slide 19 text
Represents a 1
Represents a 0
Digital Abstraction
0 V
5 V
2 V
3 V
This is wasteful - infinitely many values used to represent only two!
But useful: tolerate noise and get exact results with confidence.
Slide 20
Slide 20 text
Finite Storage
Discrete math is often infinite, but also often done over finite domains.
Slide 21
Slide 21 text
Finite Storage
Can continuous math be finite?
Slide 22
Slide 22 text
“Hard” Problems
y2 = x3 + 7
What does
y2 = x3 + 7
look like in
finite field?
Slide 23
Slide 23 text
“Hard” Problems
y2 = x3 + 7 mod G
(G = 629)
Why do we
want “hard”
problems?
Slide 24
Slide 24 text
y2 = x3 + 7 mod G
(G = 2256-232-977)
Bitcoin Market Value
Depends on it being hard to find
points on this (discrete) curve.
Slide 25
Slide 25 text
All* Discreet Communications
depend on Discrete Math
* Except for whispering and passing notes.
Me mod n
YA
=aXA mod q
Diffie-Hellman Merkle Key Agreement:
Discrete log problem is hard
RSA Public-Key Encryption:
Factoring problem is hard
Slide 26
Slide 26 text
Definitions
Slide 27
Slide 27 text
What makes a
good definition?
Slide 28
Slide 28 text
Proposition
A proposition is a statement that is either true or false.
Slide 29
Slide 29 text
Proposition
A proposition is a statement that is either true or false.
The cs2102 TAs are awesome.
2 + 2 = 5.
Today is Tuesday.
If today is Tuesday and it is during
the semester, there is cs2102 class.
Slide 30
Slide 30 text
Axioms
An axiom is a proposition that is accepted to be true.
Axioms are assumed, not proven.
Slide 31
Slide 31 text
Euclid’s Axioms
1. A straight line may be drawn between any
two points.
2. A piece of straight line may be extended
indefinitely.
3. A circle may be drawn with any given radius
and an arbitrary center.
4. All right angles are equal.
5. If a straight line crossing two straight lines
makes the interior angles on the same side
less than two right angles, the two straight
lines, if extended indefinitely, meet on that
side on which are the angles less than the
two right angles.
Slide 32
Slide 32 text
Mathematical Proof
Axioms P
Chain of steps using inference rules, starting from set of axioms, resulting in P.
Slide 33
Slide 33 text
Inference Rule: Modus Ponens
Slide 34
Slide 34 text
Inference Rule: Modus Ponens
antecedents
conclusion
Variables: P, Q
Slide 35
Slide 35 text
Soundness
An inference rule is sound, if it can only be used to
conclude true things.
antecedents
conclusion
Slide 36
Slide 36 text
Soundness
An inference rule is unsound, if there is a way to make
its antecedents all true, and its conclusion false.
antecedents
conclusion
Slide 37
Slide 37 text
Is it sound?
P
Q
Slide 38
Slide 38 text
Is it sound?
P, P implies Q
false
Slide 39
Slide 39 text
Is it sound?
P, NOT(P)
false
Slide 40
Slide 40 text
Is it sound?
NOT(P) implies NOT(Q)
Q implies P
Slide 41
Slide 41 text
Implies
Slide 42
Slide 42 text
Precise Meaning of Implies
P implies Q means: if P is true, Q must be true.
Variables are propositions: either true or false.
Slide 43
Slide 43 text
Raising Hands (Conventional)
HandRaised implies WantToSpeak
What does NOT(HandRaised) mean?
Slide 44
Slide 44 text
Raising Hands (Muller’s Rule)
NOT(WantToSpeak) implies HandRaised
What does NOT(HandRaised) mean?
Slide 45
Slide 45 text
Precise Meaning of Implies
P implies Q means: if P is true, Q must be true.
P Q P implies Q
true true true
true false false
false true true
false false true
Slide 46
Slide 46 text
Is it sound?
NOT(P) implies NOT(Q)
Q implies P
Slide 47
Slide 47 text
Is it sound?
NOT(P) implies NOT(Q)
Q implies P
P Q NOT(P) implies NOT(Q) Q implies P
true true
true false
false true
false false
Slide 48
Slide 48 text
Is it sound?
NOT(P) implies NOT(Q)
Q implies P
P Q NOT(P) implies NOT(Q) Q implies P
true true true true
true false true true
false true false false
false false true true
Slide 49
Slide 49 text
Contrapositive Rule
NOT(P) implies NOT(Q)
Q implies P
P Q NOT(P) implies NOT(Q) Q implies P
true true true true
true false true true
false true false false
false false true true
Slide 50
Slide 50 text
Inference Rule: Modus Tollens
NOT(Q), P implies Q
NOT(P)
P Q P implies Q
true true true
true false false
false true true
false false true
Slide 51
Slide 51 text
Proposition
If the product of x and y is even, at least one of x or y must be even.
Slide 52
Slide 52 text
Charge
• Before Thursday’s class: read MCS chapter 1, course
syllabus; post questions on slack
• Before Friday (6:29pm): course pledge, registration
survey (read “Habits of Highly Mathematical People”)
• I’ll have office hours now (until 4pm) and tomorrow
(Wednesday), 3-4pm [Rice 507]
• TA’s office hours will start next week (schedule to be
posted on course site)