1. Welcome to CS/MATH 320L – Applied Discrete Mathematics Spring 2015
Instructor: Marc Pomplun
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

2. Instructor – Marc Pomplun
Office: S-3-171
Lab: S-3-135
Office Hours: Tuesdays 4:00pm – 5:30pm Thursdays 7:00pm – 8:30pm
Phone: (office) (lab)
Website:
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

3. The Visual Attention Lab
Eye movement research
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

4. The new EyeLink-2K System
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

5. Example: Distribution of Visual Attention
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

6. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

7. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

8. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

9. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

10. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

11. Selectivity in Complex Scenes
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

12. Modeling of Brain Functions
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

13. Modeling of Brain Functions
unit and connection l a y e r l + 1
in the interpretive network
unit and connection
in the gating network
unit and connection
in the top-down bias network l a y e r l l a y e r l - 1
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

14. Applied Discrete Mathematics Week 1: Logic and Sets
Computer Vision:
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Applied Discrete Mathematics Week 1: Logic and Sets

15. Human-Computer Interfaces:
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Applied Discrete Mathematics Week 1: Logic and Sets

16. Applied Discrete Mathematics Week 1: Logic and Sets
Now back to CS 320L:
Course Kit:
Kenneth H. Rosen,
Discrete Mathematics and its Applications
7th Edition
(Available at the UMB Bookstore)
On the Web:
(contains all kinds of course information and also my slides in PDF and PPT formats, updated after each session)
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

17. Applied Discrete Mathematics Week 1: Logic and Sets
Your Evaluation
4 sets of exercises each set 5% % (only individual submissions allowed)
midterm (75 minutes) %
final exam (2.5 hours) %
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

18. Applied Discrete Mathematics Week 1: Logic and Sets
Grading
For the assignments, exams and your course grade, the following scheme will be used to convert percentages into letter grades:
 95%: A
 90%: A-
 86%: B+  82%: B  78%: B-
 74%: C+
 70%: C  66%: C-
 62%: D+  56%: D  50%: D-
 50%: F
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

19. Applied Discrete Mathematics Week 1: Logic and Sets
Academic Dishonesty
You are allowed to discuss problems regarding your homework with other students in the class.
However, you have to do the actual work (computing values, writing algorithms, drawing graphs, etc.) by yourself.
You cannot copy anything from other sources (Wikipedia, other students’ work, etc.)
The first violation will result in zero points for the entire homework or exam (and official notification).
The second violation will result in failing the course.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

20. Complaints about Grading
If you think that the grading of your assignment or exam was unfair,
write down your complaint (handwriting is OK),
attach it to the assignment or exam,
and give it to me or put it in my mailbox.
I will re-grade the whole exam/assignment and return it to you in class.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

21. Why Care about Discrete Math?
Digital computers are based on discrete “atoms” (bits).
Therefore, both a computer’s
structure (circuits) and
operations (execution of algorithms)
can be described by discrete math.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

22. Applied Discrete Mathematics Week 1: Logic and Sets
Syllabus
Logic and Set Theory
Functions and Sequences
Algorithms
Applications of Number Theory
Mathematical Reasoning
Counting
Probability Theory
Relations and Equivalence Relations
Graphs and Trees
Boolean Algebra
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

23. Mathematical Appetizers
Useful tools for discrete mathematics:
Logic
Set Theory
Functions
Sequences
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

24. Applied Discrete Mathematics Week 1: Logic and Sets
Crucial for mathematical reasoning
Used for designing electronic circuitry
Logic is a system based on propositions.
A proposition is a statement that is either true or false (not both).
We say that the truth value of a proposition is either true (T) or false (F).
Corresponds to 1 and 0 in digital circuits
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

25. The Statement/Proposition Game
“Elephants are bigger than mice.”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
true
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

26. The Statement/Proposition Game
“520 < 111”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

27. The Statement/Proposition Game
“y > 5”
Is this a statement?
yes
Is this a proposition? no
Its truth value depends on the value of y, but this value is not specified.
We call this type of statement a propositional function or open sentence.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

28. The Statement/Proposition Game
“Today is January 29 and 99 < 5.”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

29. The Statement/Proposition Game
“Please do not fall asleep.”
Is this a statement? no
It’s a request.
Is this a proposition? no
Only statements can be
propositions.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

30. The Statement/Proposition Game
“If elephants were red,
they could hide in cherry trees.”
Is this a statement?
yes
Is this a proposition?
yes
What is the truth value
of the proposition?
probably false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

31. The Statement/Proposition Game
“x < y if and only if y > x.”
Is this a statement?
yes
Is this a proposition?
yes
… because its truth value does not depend on specific values of x and y.
What is the truth value
of the proposition?
true
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

32. Combining Propositions
As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition.
We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

33. Logical Operators (Connectives)
We will examine the following logical operators:
Negation (NOT)
Conjunction (AND)
Disjunction (OR)
Exclusive or (XOR)
Implication (if – then)
Biconditional (if and only if)
Truth tables can be used to show how these operators can combine propositions to compound propositions.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

34. Negation (NOT) Unary Operator, Symbol:  P P true false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

35. Conjunction (AND) Binary Operator, Symbol:  P Q PQ true false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

36. Disjunction (OR) Binary Operator, Symbol:  P Q PQ true false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

37. Exclusive Or (XOR) Binary Operator, Symbol:  P Q PQ true false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

38. Implication (if - then)
Binary Operator, Symbol:  P Q
PQ
true
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

39. Biconditional (if and only if)
Binary Operator, Symbol:  P Q
PQ
true
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

40. Statements and Operators
Statements and operators can be combined in any way to form new statements. P Q P Q
(P)(Q)
true
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

41. Statements and Operations
Statements and operators can be combined in any way to form new statements. P Q
PQ
 (PQ)
(P)(Q)
true
false
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

42. Equivalent StatementsP Q
(PQ)
(P)(Q)
(PQ)(P)(Q)
true
false
The statements (PQ) and (P)(Q) are logically equivalent, because (PQ)(P)(Q) is always true.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

43. Tautologies and Contradictions
A tautology is a statement that is always true.
Examples:
R(R)
(PQ)(P)(Q)
If ST is a tautology, we write ST.
If ST is a tautology, we write ST.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

44. Tautologies and Contradictions
A contradiction is a statement that is always
false.
Examples:
R(R)
((PQ)(P)(Q))
The negation of any tautology is a contra-
diction, and the negation of any contradiction is
a tautology.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets

45. Applied Discrete Mathematics Week 1: Logic and Sets
Exercises
We already know the following tautology:
(PQ)  (P)(Q)
Nice home exercise:
Show that (PQ)  (P)(Q).
These two tautologies are known as De Morgan’s laws.
January 29, 2015
Applied Discrete Mathematics Week 1: Logic and Sets