EECS 203-1: Discrete Mathematics Winter 2005 Introductory ...

EECS 203-1: Discrete Mathematics
Winter 2005
Introductory Material
• Instructor: Prof. Bill Rounds
• Office hours: Tuesday 1:00-2:30 in 132 ATL
• GSI: Susana Early
• Office Hours: MON 5:30-7:30, FRI 2:30-3:30, 2420 EECS
• Website: UM Coursetools. The URL is
https://coursetools.ummu.umich.edu/2005/winter/eecs/203/002.nsf
• Exams: 2 midterms, outside class, Feb. 16, and Mar. 28, from 7-8:30pm,
and final exam April 21 7 – 9 PM.
• Grade distribution: HW 15%, each midterm 25%, final 35%.
• See website for other course policy.
• Another website: wwwmmhhe.com/rosen; lots of extra helpful material
on the text.
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General topics outline
• Logic and basic set theory
• Elementary methods of proof
• Induction
• MT 1
• Counting
• Relations
• MT 2
• Graphs, algorithms, and complexity
• See website for specific syllabus.
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Part I: Logic
(Text section 1.1)
• We’ll begin by studying propositional logic.
• This is a simple system useful in translating informal statements into precise
mathematical form.
• In a slightly different notation, propositional logic is used to design logic
circuits, also called Boolean circuits.
• George Boole (1815-1864) is sometimes called the father of modern logic.
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Propositions
• A proposition is a declarative statement that has a definite truth content:
either true or false.
• Examples: (i) Toronto is the capital of Canada. (ii) 2 + 2 = 3. (iii) Kevin
Bacon is the Pope. (iv) Saddam is the new U of M assistant defensive
coordinator.
• Non- propositions: (i) What time is it (ii) x + 1 = 4. (iii) He is the Pope.
• Sometimes you have contextual information that tells you what x’s value
is, or who ”he” is. Then statements like (ii) and (iii) are propositions, but
only relative to a context or situation.
• A statement like x + x = 2x is technically not a proposition, even though
it’s true for all numbers x.
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Compound propositions
• These are propositions that are built out of other ones, starting with basic
ones, using logical operators, also called propositional connectives or
Boolean connectives.
• We start with a simple operator called negation.
Definition Let p be a proposition. The statement
“It is not the case that p”
is called the negation of p, and is denoted by ¬p, and read “not p”.
Example (i) (Relative to Jan. 7, 2005) Let p be “Today is Tuesday”. Then
¬p is “It is not the case that today is Tuesday”, or “today is not Tuesday”, or
“it is not Tuesday today”, (ii) Let p be “1 + 1 = 2” Then ¬p is “1 + 1 ≠ 2”.
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Other logical operators and truth tables
• You can take the negation of any proposition, whether or not it is true or
false. The result will be a new proposition with the opposite truth value.
• We express this using a simple display called a truth table.
• The truth table for negation is
p ¬p
T F
F T
• We now introduce two new operators called conjunction and disjunction.
(These are also called “and” and “or”.)
• Examples (relative to today) p is “it is raining today” and q is “today is
Friday”. The proposition “p and q”, notated p ∧ q, is “it is raining today
and today is Friday”. The proposition “p or q, notated p∨q, is “it is raining
today or today is Friday.”
• The truth tables for conjunction and disjunction:
p q p ∧ q
T T T
T F F
F T F
F F F
p q p ∨ q
T T T
T F T
F T T
F F F
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Still more operators
• The exclusive or (⊕) operator has the following truth table:
p q p ⊕ q
T T F
T F T
F T T
F F F
You can see from that p and q have to have differing truth values in order
for p ⊕ q to be true.
• There is also an operator requiring p and q to have the same truth value.
This is the biconditional operator p ↔ q:
p q p ↔ q
T T T
T F F
F T F
F F T
• p ↔ q is read “p if and only if q”. It’s also read as ”p is necessary and
sufficient for q” and sometimes as ”p iff q” (note the two f’s).
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The conditional connective
• One connective that gets a big workout in logic is the conditional p → q.
It is read “if p then q”, ”p is sufficient for q, ”a necessary condition for p is
q”, “p only if q”, and ”q whenever p”.
• You can think of this as a “one-way biconditional”. The truth table is
p q p → q
T T T
T F F
F T T
F F T
• One way to read this table is that ”q is at least as true as p is, and maybe
more,” as in the third line. Another way to read it is ”p can’t be true and q
false.” This has some strange repercussions. For example,
is true (by line 4). Equally,
is true.
if 2 + 2 = 5, then Kevin Bacon is the Pope
if 2 + 2 = 5, then Kevin Bacon is not the Pope
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Complex expressions in propositional logic
• You can form “algebraic” expressions using these connectives. For example,
(¬p ↔ ¬q) ↔ (¬p → (p ∨ q))
• There is a precedence scheme here, like the precedence scheme for addition,
multiplication, and negation in algebraic expressions.
• You can build truth tables for complex expressions. For example, consider
We build the table this way:
p → (¬q ∨ r).
p q r ¬q (¬q ∨ r) p → (¬q ∨ r)
T T T F T T
T T F F F F
T F T T T T
T F F T T T
F T T F T T
F T F F F T
F F T T T T
F F F T T T
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Translating English to propositional logic
• Because English has compound propositions, it’s sometimes possible to
phrase these propositions as formal propositional expressions and to build
truth tables therefrom. This is a bit inexact, and there is very often more
than one way to do it. There’s a way to test whether two translations are
equivalent, though.
• Example from text: “You cannot ride the roller coaster if you are under 4
feet tall unless you are older than 16.”
• We find the basic propositions in this informal expression and assign them
propositional variables: “you can ride the roller coaster” is assigned r,
”you are under 4 feet tall” is assigned u, and ”you are over 16” is assigned
o. Then the English sentence before ”unless” would translate as ”u → ¬r”.
• But what about the “unless” part Here o qualifies u in the sense that if
somebody is under 6 feet, and not over 16, then they cannot ride. So a
correct translation is
(u ∧ ¬o) → ¬r.
• How about (u → ¬r) ∧ (o → r) Is this the same What does “same”
mean To be continued in the next lecture.
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