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Lect 13 - November 6, 2008 - ECS 20 - Fall 2008 - Phil Rogaway
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Today: o Review
o Infinite Sets (done well in Velleman Chapter 7)
o Return, comment on your midterms
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1. Review of big-O, Theta
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Special functions
\lceil .. \rceil
log(ab) = log(a) + log(b)
log_a(b) = log_c(b) / log_c(a)
e^ab = (e^a)^b
a^x a^y = a^{x+y}
Asymptotics
O(g) = {f: \N -> \R: \exists C, N s.t.
f(n) <= C g(n) for all n>= N}
\Theta(g) = {f: \N -> \R: \exists c,C N s.t.
c g(n) <= f(n) <= C g(n) for all n>= N}
Example:
True/False:
If f is Theta(n^2) then f is O(n^2) TRUE
n! = O(2^n) NO
n! = O(n^n) YES
(Truth: n! = Theta((n/e)^n sqrt(n)))
Claim: H_n = 1/1 + 1/2 + ... + 1/n = O(lg n)
Draw picture.
Upperbound by 1 + \integral_1^n (1/x)dx = 1 + ln(n) = O(lg n)
show Could I write O(log n)
Sure; O(log n) = O(lg n)
How many steps are sufficient to solve the Towers of Hanoi problem:
"I don't recall exactly, but it's O(2^n)."
Compute the asymptotic running time of the following algorithm:
Given: a formula phi, decide if phi is satisfiable
by trying all possible truth assignments.
Suppose phi has n variables and takes m bits to write down.
Answer: Need to try 2^n t.a. How long to try teach? O(m).
All together? O(m 2^n).
One reason asymptotic notation handy:
O(n^2) + O(n^2) = O(n^2)
O(n^2) + O(n^3) = O(n^3)
O(n log n) + O(n) = O(n log n)
etc.
A reason we use asymptotic notation: model independence
How long does the following code take to run
for i=1 to n do
for j=1 to i do
s += (i+j)^2 - (i+j)
How long to sort by bubble sort
O(n^2)
Already enough to distinguish from more sophisticated algorithms.
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2. How big is that infinity?
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Puzzle: Which are there more of: natural number or integers?
Mathematicians answer: they are the same. Not because they
are both infinite, but because there is a bijective function
f: \N -> \Z
e.g,
f(x) = \lceil x/2 \rceil if x is odd
-x/2 if x is even
Is there a bijective function from \Z to \N
Answer 1: yes, automatic from above
Answer 2, yes, f(x) = -2x if x>0
2x-1 if x is odd
-3 -2 -1 0 1 2 3
0
Woops! -- student told after class that both I and the book
had earlier defined \N not to include 0.
In fact, both conventions are common, but
I did not mean to vascillate.
Puzzle: is there a bijective function from \N to \N x \N
Technique: "Dovetailing" // Did not get to in lecture, Tung will do
// this in discussion section
Definition: Let A and B be sets. We say that A is *equinumerous* to B,
A ~ B, if there is a bijection f from A to B.
(Woops! In class I said "equicardinal", but Velleman uses "equinumerous")
Definitions:
A set is *finite* if it is equinumerous to I_n = {1,...,n} for some n\in \N.
A set is *infinite* if it is not finite.
A set is *countably infinite* it is equi numerable to \N
A set is *countable* if it is finite or countably infinite.
A set is *uncountable* if it is not countable.
Proposition: ~ is an equivalence relation
Proposition: \Q is countably infinite.
Proposition: if A and B are countable then so is A x B and A u B
Theorem: P(\N) is uncountable. (the powerset of the natural numbers)
Proof. Suppose for contradiction that P(\N) were countable:
say P(\N) = \{A_1, A_2, ...}
We construct a subset B of \N as follows
B contains i\in iff i \not\in A_i
Now B is a subset of \N so it must be in the enumeration: say B = A_j
But: is j \in B?
j \in B iff j\not\in A_j = B
Contradiction.
Theorem: \R is uncountable.
Prove it in the usual way.
Proposition: (0,1) ~ \R (not done in class)
0 1
<-------------(-------)------->
f(x) = 2(x-1/3) = 2x-1
-1 0 1
<------(-------------)------->
g(x) = 1/(1-x) - 1 if x>=0
g(x) = 1/(1+x) - 1 if x<0
<-------------|--------------->
Definition: For sets A, B, write A \le B if there exists a one-to-one function
f: A -> B.
Theorem: [ Cantor-Schr\"oder-Bernstein ] [not done in class]
If A \le B and B \le A then A ~ B
Corollary: \R ~ P(\N)
Proofs: See your book!
Names (kind of like numbers) for different size infinite
\Aleph_0
c for the "size" of \R: |\R| = c
Continuum Hypothesis: 1900, David Hilbert's first (of 23) problem.
1963 - Paul Cohen proved not provable under Zermelo-Frankel
set theory with choice.
There is no set with cardinality strictly between \Aleph_0 and c:
Cannot be proven nor disproven under in ZFC!
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3. Return and comment on quizzes
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