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EECS 210
Instructor: Dr. Nancy Kinnersley
Office: 2040 Eaton
Phone: 864-7390
email: [email protected]
web page:
http://people.eecs.ku.edu/~kinners/210_s11_index.html
TA: Yi Jia 2041 Eaton
Office Hours:
Text: Discrete Mathematics—Mathematical Reasoning
and Proof with Puzzles, Patterns, and Games by Ensley
and Crawley
Homework:
Exams and Quizzes:
Final Exam:
Discussion Groups:
Class Policies:
Grading:
One of four siblings broke a vase. Knowing that only one of the
children tells the truth, which of the following statements is true?
Who broke the vase?
Betty: Steve broke the vase.
Steve: John broke it.
Laura: I didn't break it.
John: Steve lied when he said I broke it.
A light bulb is hanging in a room. In another room there
are three switches of which only one will turn the light
on. Initially all switches are "off" and the bulb is not lit.
How can you determine which switch lights the bulb if
you are only allowed to enter the room with the bulb one
time?
Section 1.2 Number Puzzles and Sequences
sequence
pattern
1, 4, 9, 16, 25, …
0, 5, 10, 15, …
1, 3, 9, 27, …
1, 2, 6, 24, 120 …
Defn: A recursive formula for a sequence
A closed formula
A geometric progression is
Notational conventions for sequences
closed form
Other recursive patterns and sequences
Fibonacci numbers
Summations
Sums you are expected to know:
n
i = 1 + 2 + 3 + … + n = (n)(n+1)/2
i=1
n
 (2i - 1) = 1 + 3 + 5 + … + 2n - 1 = n2
i=1
n
2
i = 12 + 22 + 32 + … + n2 = (n)(n+1)(2n+1)/6
i=1
Using Summation Formulas
5
 2•3i =
i=0
10
 2i =
i=0
10
 2i =
i=6
Summing a geometric progression
Algebraic proof of the formula for the sum of a geometric
progression
Section 1.3 Truthtellers, liars and propositional logic
We call a sentence a proposition
p:
q:
r:
A formal proposition
1.
2. Given propositions p and q, their conjunction, written p  q is
3. Given propositions p and q, their disjunction, written p V q
4. Given a proposition p, the compound statement p
Truth Tables
bit operations corresponding to logical connectives
1010 1100
0011 1011
AND
OR
XOR
Combining operations
Two statements are logically equivalent
Proposition 1 (p. 32) (DeMorgan’s Laws) Let p and q be any
propositions. Then
1. (p V q) is logically equivalent to p  q
2. (p  q) is logically equivalent to p V q
Example
A tautology is
A contradiction is
Table of Equivalences
Commutative
Associative
Distributive
Identity
Negation
Double negative
Idempotent
DeMorgan's Laws
Universal
Absorption
pqqp
(p  q)  r = p  (q  r)
p  (q V r)  (p  q) V (p  r)
pTp
p V p  T
 
( p)  p
ppp

(p  q)  p V q
pVTT
p  (p V q)  p
pVqqVp
(p V q) V r  p V (q V r)
p V (q  r)  (p V q)  (p V r)
pVFp
p  p  F
pVpp

(p V q)  p  q
pFF
p V (p  q)  p
Proof Example
Show (p  q) V q  p V q
Section 1.5 Implications
An implication is
Truth table for the implication
p
q
T
T
T
F
F
T
F
F
pq
The biconditional statement:
Notation: p  q.
p
q
T
T
T
F
F
T
F
F
pq
Terminology and Precedence
Alternate Ways of Expressing Implications
English Word
Logical
Connective
Logical
Expression
and; but; also; moreover
conjunction
pq
or
disjunction
pVq
implication
pq
if p, then q
p implies q
p, therefore q
p only if q
q follows from p
p is a sufficient condition for q
q is a necessary condition for p
p if and only if q
equivalence
pq
p is necessary and sufficient for (biconditional)
q
not p
it is false that p …
it is not true that p …
negation

p
Examples:
The flowers bloom only if you water them.
If it rains hard, it’s necessary to carry and umbrella.
Having one even factor is sufficient to have an even product.
Contrapositives, converses, and inverses
Definition Consider the implication p  q
1. The converse of the implication is
2. The inverse of the implication is
3. The contrapositive of the implication is
Proposition 3
1. An implication and its contrapositive are logically equivalent
2. The converse and inverse of an implication are logically
equivalent
3. The implication is not logically equivalent to its converse
Showing p  q and q  p are logically equivalent and that p  q
and q  p are not
p
q
q
p
pq
q  p
qp
Using Equivalence in Proofs
Statement: If n2 is odd then n is odd
Contrapositive:
Indirect (contrapositive) proof:
Direct proof:
Statement: If a and b are odd numbers, then a•b is odd.
Inverse:
Translating English sentences
If the file is not damaged and the processor is fast, then the
printer is slow.
Translate: The program is efficient only if it executes fast and
doesn’t have a bug.
p:
q:
r:
System Specifications
Consistency
Consistency Example
1. The system is in multiuser state iff it is operating normally.
2. If the system is operating normally, the kernel is functioning.
3. The kernel is not functioning or the system is in interrupt mode.
4. If the system is not in multiuser state then it is in interrupt mode.
5. The system is not in interrupt mode.
p:
q:
r:
s: