Name _ Discrete 131
Assignment #10
1.
Review for 3rd Unit Exam
How many different 2-digit numerals can be made from the digits 3, 4, and 5 if a
digit can appear just once in the numeral?
3
P2 = 6
2. Six people enter a room with 4 empty chairs in a row. In

Notes on Intersection, Union and Complements
The set of all elements that belong to both A and B is called the
intersection of sets A and B, denoted A B.
The set of all elements that belong to either A or B (or both) is
called the union of sets A and B, d

Quantifiers
Rewrite each statement formally using quantifiers and variables.
1. All triangles have three sides.
triangles
t, t has three sides or t T, t has three sides
( where T is the set of all triangles)
2. No dogs have wings.
dogs d, d does not have

The Quotient-Remainder Theorem
Given an integer n and positive integer d,
there exists unique integers q and r such that:
n = dq + r and 0 r < d
ex. 1.
n = 62, d = 3
62 = 3 * 20 + 2 hence q = 20 and r = 2
2.
n = -33 d = 5
-33 = (-7) * (5) + 2 hence q = -7

Notes on Sets
Sets
A well-defined collection of objects is called a set.
An object which belongs to a set is said to be a member or
element of the set.
Well-defined:
The set of all U.S. presidents.
The set of all even integers greater than 100.
The set of

Statements
A statement is a declarative sentence that is either true or false.
Which of the following are statements?
This is Discrete Math 205.
2x + y = 17
The president of the United States is Bill Clinton.
9 - 7 = 2
Logical statements are joined by con

11.5 Spanning Trees
A spanning tree for a graph G is a subgraph of G that contains every
vertex of G and is a tree.
1. Every connected graph has a spanning tree.
2. Any two spanning trees for a graph have the same number of
edges.
A weighted graph is a gr

11.5 Trees
A graph is said to be circuit-free if, and only if, it has no nontrivial
circuits. A graph is called a tree if, and only if, it is circuit-free and
connected. A trivial tree is a graph that consists of a single vertex and an
empty tree is a tre

Notes on 1.3
Valid and Invalid Arguments
Premises are also called assumptions or hypotheses
Conclusions are final statements.
If you have true premises and a false conclusion, then the argument
is invalid.
Modus Ponens in Latin means method of affirming
p

Quantifiers
We defined sets by specifying a property P ( x ) that
elements of the set have in common. Thus, an element
cfw_x / P ( x ) is an object x for which the statement P (x)
is true. Such a sentence P ( x ) is called a predicate,
because in English

Notes on 7.3
Application: The Pigeonhole Principle
A function from one finite set to a smaller finite set cannot be oneto-one.
There must be at least two elements in the domain that have the
same image in the co-domain.
Problems:
1. A small town has 560 r

11. 4 Isomorphisms of Graphs
Let G and G be graphs with vertex sets V(G) and V(G) and edge sets
E(G) and E(G), respectively. G is isomorphic to G if, and only if, there
exists one-to-one correspondence g: V(G) V(G) and h: E(G)
E(G) that preservs the edge

The Irrationality of 2
To find the diagonal of a square with sides of 1, use the
Pythagorean Theorem.
c 2 = a 2 + b2
c 2 = 1 2 + 12
c2 = 2
c = 2
The ancient Greeks believed any 2 given line segments
A: _ and B: _,
There are two integers a and b that can b

Notes on 1.1
Logical Equivalences
Given any statement variables p, q, and r, a tautology t and
a contradiction c, the following logical equivalences hold.
1. Commutative
p
q q
2. Associative
(p q) r p (q
r)
3. Distributive
p (qvr) (p q)v(p r)
4. Identity

Notes on Matrices
Matrices is the plural of matrix
A matrix is a rectangular array of data.
Each row and column must have a label.
Each piece of data in the matrix is called an entry.
The dimension of a matrix is the number of rows by the number of
column

11.3 Matrix Representation of Graphs
An m x n matrix A over a set S is a rectangular array of elements
of S arranged into m rows and n columns.
A matrix for which the numbers of rows and columns are equal is
called a square matrix.
The main diagonal consi

Necessary and Sufficient Conditions, Only if
1. x, r(x) is a sufficient condition for s(x) means x, if r(x) then s(x).
2. x, r(x) is a necessary condition for s(x) means x, if ~ r(x) then ~ s(x) or
equivalently, x, if s(x) then r(x).
3. x, r(x) only if s(

Notes on 1.2 Conditional Statements
Negation of a conditional
~ ( p q) p
~q
If I studied for the quiz, I should get an A.
Negation: I studied for the quiz and I did not get an A.
Truth Table
p
q
p q
~(pq)
~q
p
~q
Converse: Change the hypothesis to the con

11.2 Paths and Circuits
Let G be a graph and let v and w be vertices in G.
A walk from v to w is a finite alternating sequence of adjacent
vertices and edges of G.
v0e1v1e2 . . . vn 1 envn
A trivial walk from v to w consists of the single vertex v.
A path

Notes on 6.2 Possibility Trees and Multiplication Rule
Probability is the chance or likelihood of a particular event
occurring.
Probability is a number 0 p 1.
Tournament
2 teams A and B
To win:
1. You must win two games in a row
2. You must win three game

Name _ Practice Final Exam 205 Fall 2010
Part I
1. Let U = cfw_1, 5, 7, 8, 9
Which of the following is true about the relation
R = cfw_(1, 5), (7, 8), (5, 5), (9, 7), (7,7), (8, 8)?
a.
The relation is a function.
b. The relation is not a function because

Name _
Review for Quiz on binary, congruence, gcd, lcm
1.
Convert 623 to binary.
2.
Convert 10111101012 to decimal.
3.
Add: 10011101012 + 1011110112
4. Subtract: 1010001112 - 101110112
5.
Find the gcd of (321, 5457) Show your work!
6.
Find the lcm of (36,

Vectors, n-tuples, lists
A vector is a finite sequence of entries for instance,
<2, 4, -8, 5 >. Tuples, <x, y>, with entries which are real numbers
are used to describe points of the plane. Triples, <x, y, z>, with
entries which are real numbers to descri

Name _
Review of Integer Algorithms, Venn Diagrams and Congruence
1. Does 106 0 ?
2. Find integers q and r such that n = dq + r and 0 r < d.
a. n = 62 and d = 8
b. n = -26 and d = 6
c. n = 11 and d = 16
3. When an integer a is divided by 7 the remainder i

Review for unit exam (Chapters 5 & 6)
1. Draw a Venn diagram describing each of the following by shading in the
appropriate group of regions.
a. A U (B C)
b. (A B)
2. Verify using a Venn diagram proof: (A B) = A U B
3. A survey of 200 people was conducted

Answers for Review for Unit exam # 9 20
9.
3
P2 =3x2=6
10. 6 P 4 = 6 x 5 x 4 x 3 = 360
11. 11! / 3! 2! 3! = 554,400
12.
C 4 = 210
C 2 x 4 C 2 = 15 x 6 = 90
6
x 4C 4 = 1
6C 0
C 4 x 4 C 0 = 15
6
10
13. 3 x 2 x 1
15!
14. 2 C 1
x
= 6
2
=
C1
x 3C 2
3
P3
= 2 x

Discrete Math 205 Review for 1st Unit exam Winter 2005
Review of the binary, octal and hexadecimal systems.
1.
Convert 385 to binary.
2.
Convert 111010112 to decimal.
3.
Add: 11111102 + 1111012
4. Subtract: 11000012 1101112
5. Convert the following decima

Name _ Date _
Review sheets for Unit Exam on Chapters 2 & 3 & Data Compression and RSA
Encryption
1. Does 10 0 ?
2. Find integers q and r such that n = dq + r and 0 r < d.
a. n = 59 and d = 7
b. n = -14 and d = 6
c. n = 12 and d = 26
3. When an integer a