Unformatted text preview: xists a constant L and a natural number n0 such
that
L g(n) <= f(n)
holds for all n >= n0.
In other words, f(n) = Ω(g(n)) if and only if g(n) = O(f(n)). 27 Big Θ
We define f(n) = Θ(g(n)) if and only if there exist constants L
and U and a natural number n0 such that
Lg(n) <= f(n) <= Ug(n)
holds for all n >= n0. In other words, f(n) = Θ(g(n)) if and only if
f(n) = Ω(g(n)) and f(n) = O(g(n)). 28 Counting
You need to know
 the basic counting principles (product rule, sum rule,...)
 (generalized) pigeonhole principle
 permutations and combinations
 binomial coefficients
 binomial theorem
 counting with repeated elements 29 Pigeonhole Principle
In any cocktail party n >=2 people, there must be at least two
people who have the same number of friends (assuming that the
friends relation is symmetric and irreflexive).
The number of friends of each person ranges between 0 and n1.
Case 1: Everyone has at least one friend.
If everyone has at least one friend, then each person has between 1 to n1
friends. Since we have n people, and just n1 different values, there must be
two partygoers that have the same number of friends by the pigeonhole
principle.
Case 2: Someone has no friends.
If someone lacks any friends, then that person is a stranger to all other
guests. Because friend is symmetric, the highest value anyone else could
have is n  2, that is, everyone has between 0 to n – 2 friends. Since we have
n people, and just n1 different values, there must be two partygoers that
have the same number of friends by the pigeonhole principle. Solving Recurrences You need to be able to solve recurrences by
 solving characteristic equations (for homogeneous linear
recurrences of degree 2)
 by inspecting, guessing, and verifying a solution
 by applying the master theorem 31 Relations
You need to know
 the basic properties of relations (reflexive, symmetric,
antisymmetric, transitive, ...)
 how to show that a relation is an equivalence relation
 the congruence relation mod m
 how to show that a relation is a partial order
 what an order lattice is 32 Formal Languages
You need to
 be familiar with the Chomsky Hierarchy
 be able to determine the language associated with a
grammar
 know the characterization of regular languages as
languages accepted by finite state automata or languages
described by regular expressions
 be familiar with finite state machines and finite state
automata 33 Hints Read the textbook and the class notes.
Study old exams, quizzes, homeworks.
Drill using odd numbered exercises.
Scan through review sections of the textbook. Get enough sleep!!
34...
View
Full Document
 Fall '11
 math
 Logic, truth values, binary function MX, n1 different values

Click to edit the document details