M IDTERM EXAM INATION Nathan Redford CSC 226-002 10/14/2008
19.
If we align one set across the top of the diagram and the other down the left hand side of the diagram, we can list all cartesian products at the intersections of the sets. We can then
Homework 1- CSC 224/226
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WebAssign Homework 1- CSC 224/226 (Homework) Current Score: 117.625 out of 117.625 Due: Tuesday, August 26, 2008 11:59 PM EDT Description Read the question and instructions care
This le contains the exercises, hints, and solutions for Chapter 2 of the book Introduction to the Design and Analysis of Algorithms, 2nd edition, by A. Levitin. The problems that might be challenging for at least some students are marked by ; those that
1. P T T T T F F F F
a. p (q r) (p q) (p r) q T T F F T T F F r T F T F T F T F q r T T T F T T T F p (q r) T T T F F F F F p q T T F F F F F F p r T F T F F F F F (p q) (p r) T T T F F F F F
b. (q r) p (q p) (r p) P T T T T F F F
Special Test 1
Topic: Propositional Calculus and Truth Tables
1.
p (q r )
(q s) t
(p s)
Prove t using a direct proof.
2.
pq
rs
(p s)
Prove (q r) using proof by contradiction.
Given:
Given:
p (q r )
rs
(q s)
Prove:
p
Fill in the blanks for the following pr
CSC 224/226 Discrete Mathematics
First Exam Review
Write your name on all sheets.
Reasons given in proofs must include line numbers as well as rule.
Rules can be given by name or number.
THIS IS A CLOSED-BOOK, CLOSED NOTE EXAM (except for the Axiom List).
Jason Krontz 3-9-09 CSC 226 MIDTERM 1.)
p T T T T F F a.) F (q T T F F T T F b.) F
(q v r) (p q) v (p r)
T T T T T F F T T F F T T F T F F F T F F T F T F T F T T T T T T T T T T T T F F F T T T T T T F T T F F T T F T T T T T F F T T T T F F F
University of Washington, Tacoma TCSS 342, Winter 2006, Section B (Hong) Assignment #7 version 1.1 Due: Thursday, March 9, 2006, 4:15 PM Please show your work and explain your reasoning. Write legibly (or type) and organize your answers. Turn in your own
CSC 226 Course Syllabus
CSC 226 Discrete Mathematics for Computer Scientists
Sections 001 & 002
SPRING 2017
3 Credit Hours
Course Description
CSC
226
Discrete Mathematics for
Computer Scientists
UNITS:
3 - Offered in Fall
and Spring
Prerequisite: MA 101 o
L1.5-7
Lecture Notes: Methods of Proof
All proofs are alike!
This sounds like a broad and capricious generalization but it is nonetheless
true. Any proof begins with axioms or premises, (or whatever you wish
to call them), which are (assumed to be) true p
Bipartite Graphs and Subgraphs
A bipartite graph is a simple graph whose
vertex set can be partitioned into two sets. Edges
may only connect vertices in different sets.
E.g., The men and women in a two-way
polygamous but purely heterosexual village.
Marri
About Integers:
Division algorithm:
Let a be an integer and d be a positive integer.
Then there are unique integers q and r such that
a = dq + r, 0 r<d.
7 3=2 with remainder 1: a=7, d=3, q=2, r=1.
-7 3=-3 with remainder 2: a=-7, d=3, q= -3, r=2
7 -3 is tr
L9.2-3
Lecture Notes: Bipartite Graphs, Subgraphs, Representing Graphs and Graph
Isomorphisms
A bipartite graph is a simple graph whose vertex set can be partitioned into
two sets such that the end points of any edge are in different sets.
In effect, then
Consider |AB| when |A| and |B| are not
necessarily disjoint.
A = cfw_a,b,c,d,e,f,g B = cfw_e,f,g,h,i,j.
AB = cfw_a,b,c,d,e,f,g,e,f,g,h,i,j =
cfw_a,b,c,d,e,f,g,h,i,j.
AB = cfw_e,f,g
|AB| = |A| + |B| - |AB|.
This is the first and simplest inclusionexclusion
Permutation: A way of ordering elements in a
set
E.g., Let S = cfw_a,b,c. Then bca is a permutation
of S, as is abc, etc. Permutations do not have
repeated elements.
Let S be a set of n elements. An r-permutation
of S is a way of ordering r elements of th
L5.3-5
Lecture Notes: Permutations and Combinations
Permutation: A way of ordering elements in a set. (Recall that the elements
of a set are inherently unordered never mind the fact that we are forced to
write them in order.)
E.g., Let A = cfw_a,b,c. Then
L4.1-2
Lecture Notes: Mathematical Induction
It is often the case that we want to consider the truth of statements such as
(1)
n
i = n(n+1)/2, n0
i =0
This means that the sum from zero to any natural number is given by the
formula. This is another instanc
Review of Sequences:
-A sequence is a function from N or a subset of
N to a set. It is finite if the subset of N is finite.
-Finite sequences are considered strings.
-For example 1,1/2,1/4,1/8,1/16, . . is a sequence
where a0=1, a1=1/2, a2 = 1/4, etc.
-Se
L7.1-2
Lecture Notes: Recurrence Relations
Review of Sequences:
-A sequence is a function from N (the natural numbers), or a subset of N, to
a set. It is denoted generically as a0, a1, . . an . . and always begins with the
zero index unless specifically n
L3.3
Lecture Notes: Algorithms and their complexity
One of the reasons for learning about the bounding of functions is for
algorithm analysis. We would like to examine an algorithm, come up with a
function that approximates its running time when given an
L3.5-6
Lecture Notes: Primes and Greatest Common Divisors (GCDs)
An extremely important concept is that of the prime number. Prime numbers
are to mathematics what atoms are to chemistry: the building blocks of
everything.
A positive integer p > 1 is prime
ITCS2175-Logic and Algorithms
Second Hour Exam REVIEW
Set Theory, Predicate Calculus, Arithmetic Proofs
Induction, Recursion
75 minutes
Write your name on all sheets.
Reasons given in proofs must include line numbers as well as rule.
Rules can be given by
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ITCS2175-Logic and Algorithms
Second Hour Exam REVIEW
Set Theory, Predicate Calculus, Arithmetic Proofs
Induction, Recursion
75 minutes
Write your name on all sheets.
Reasons given in proofs must include line numbers as well as rule.
Rules can be given by
V5 Homework? x a
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5(2") = 25(1) 1- 2k
Find the sequence 5(2 ) for k - u,1,2,3,4,5,s
Fill in the rst box with the terms you emulate fnr k in the farm:
2*(
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Prove ZAn > nAZ by Induction using a basis ) 4:
E32 >25
HE " HE
Mme:
_ HE HE
2"(+1) 2' E (n+1) -or n+1
Prove:
E
mum-s 2._EE >EE