Math/CSE 1019N:
Discrete Mathematics for Computer Science
Winter 2007
Suprakash Datta
datta@cs.yorku.ca
Office: CSEB 3043
Phone: 416-736-2100 ext 77875
Course page: http://www.cs.yorku.ca/course/1019
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1
Administrivia
Textbook:
Lectures: Tu-Th 10:00-11:30 am (CLH E)
Exams: 2 tests (40%), final (40%)
Homework (20%): equally divided
between 3 or 4 assignments.
Slides: should be available the previous
evening
Office hours: Tuesday 12-2 pm, Wed 1-2
pm or by appointment at CSEB 3043.
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Kenneth H. Rosen.
Discrete Mathematics
and Its Applications,
Sixth Edition. McGraw
Hill, 2007.
2
Administrivia – contd.
• Cheating will not be tolerated. Visit the
webpage for more details on policies.
• TA: Tutorials/office hours TBA.
• I will have some extra-credit quizzes.
These will be announced beforehand.
• HW submitted late will not be graded.
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Course objectives
We will focus on two major goals:
• Basic tools and techniques in discrete
mathematics
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Propositional logic
Set Theory
Simple algorithms
Induction, recursion
Counting techniques (Combinatorics)
Introductory graph theory
• Precise and rigorous mathematical reasoning
– Writing proofs
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To do well you should:
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•
•
•
•
•
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Study with pen and paper
Ask for help immediately
Practice, practice, practice…
Follow along in class rather than take notes
Ask questions in class
Keep up with the class
Read the book, not just the slides
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Today
Ch. 1: Introduction to Propositional Logic
• Truth values, truth tables
• Boolean logic:   
• Implications:  
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Why study propositional logic?
• A formal mathematical “language” for
precise reasoning.
• Start with propositions.
• Add other constructs like negation,
conjunction, disjunction, implication etc.
• All of these are based on ideas we use
daily to reason about things.
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Propositions
• Declarative sentence
• Must be either True or False.
Propositions:
• York University is in Toronto
• York University is in downtown Toronto
• All students at York are Computer Sc. majors.
Not propositions:
• Do you like this class?
• There are x students in this class.
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8
Propositions - 2
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•
•
•
•
Truth value: True or False
Variables: p,q,r,s,…
Negation:
p (“not p”)
Truth tables
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p
p
T
F
F
T
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Conjunction, Disjunction
• Conjunction: p  q [“and”]
• Disjunction: p  q [“or”]
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p
q
p q
pq
T
T
T
T
T
F
F
T
F
T
F
T
F
F
F
F
10
Exclusive OR (XOR)
• p  q – T if p and q have different truth
values, F otherwise
• Colloquially, we often use OR
ambiguously – “an entrée comes with
soup or salad” implies XOR, but
“students can take MATH XXXX if they
gave taken MATH 2320 or MATH 1019”
usually means the normal OR (so a
student who has taken both is still
eligible for MATH XXXX).
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11
Conditional
• p  q [“if p then q”]
• p: hypothesis, q: conclusion
• E.g.: “If you turn in a homework late, it will not
be graded”; “If you get 100% in this course,
you will get an A+”.
• TRICKY: Is p  q TRUE if p is FALSE?
YES!!
• Think of “If you get 100% in this course, you
will get an A+” as a promise – is the promise
violated if someone gets 50% and does not
receive an A+?
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Conditional - 2
• p  q [“if p then q”]
• Truth table:
p
q
p q
pq
T
T
T
T
T
F
F
F
F
T
T
T
F
F
T
T
Note the truth table of  p  q
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Logical Equivalence
• p  q and  p  q are logically equivalent
• Truth tables are the simplest way to
prove such facts.
• We will learn other ways later.
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14
Contrapositive
• Contrapositive of p  q is q  p
• Any conditional and its contrapositive
are logically equivalent (have the same
truth table) – Check by writing down the
truth table.
• E.g. The contrapositive of “If you get
100% in this course, you will get an A+”
is “If you do not get an A+ in this course,
you did not get 100%”.
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Converse
• Converse of p  q is q  p
• Not logically equivalent to conditional
• Ex 1: “If you get 100% in this course,
you will get an A+” and “If you get an A+
in this course, you scored 100%” are
not equivalent.
• Ex 2: If you won the lottery, you are rich.
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Other conditionals
Inverse:
• inverse of p  q is p  q
• How is this related to the converse?
Biconditional:
• “If and only if”
• True if p,q have same truth values, flase
otherwise. Q: How is this related to XOR?
• Can also be defined as (p  q)  (q  p)
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Readings and notes
• Read pages 1-12.
• Think about the notion of truth
tables.
• Master the rationale behind the
definition of conditionals.
• Practice translating English
sentences to propositional logic
statements.
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