COMP SCI 241 Discrete Mathematics I
Practice number system conversions
Fill in the following table so that each rows entries are equal in value, but represented in the positional
number system named at the top
x
2W , c How to use this recursive denition:
Denition ( a long one): Given a square matrix A, the elf O; d Jwpick any convenient row of A.
determinant of A, denoted |A|is the scalar whose value is (g
computed as follows: {i r /*b g' ' For each ent
i. >ary Counting Techniques.
Multiplication Principle: If an object can be constructed int
successive steps, where
step 1 can be performed in n1 different ways, and
step 2 can be done in n; different ways (regardless of the
choice made in step 1), and
st
)
Some denitions from Number Theory:
Aninteger a is divisible by an integer b, written bla if there
exists an integer e such that bc=a.'
If a is divisible by b, we say a is a multiple of b and b is a factor
or divisor of a.
Examples: .
7|21 because 7(3
If. i ,' . ' f" ' ' u y
, g 3 , _
h L) a Computation W6 W111 need In a mathematical " ' , ' Heres an example from Discrete Mathematics-.11:
induction proof later in the course: '
102<k+1>1 =-102k+21=102<1+2=102-102k1 i i . . Show thatiff(n) is the functi
Matrices
A matrix is a rectangular arrangement of objects
called element's (Or entries);
The entries are usually real numbers, and when they
are, the matrix is a real matrix. '
Sometimes the entries are variables 'or functions.
Sometimes the entries are 0
.L_ogi_c
The purpose of studying logic is to determine if our reasoning is
correct. It is not concerned with determining if a statement is
true or false.
There is a big difference betWeen mathematical argument and
rhetorical argument: the former is concer
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) Positional Number Systems.
The concept of the "counting numbers or natural
numbers" is one of the oldest and most basic of
mathematical concepts.
As mathematics developed, the need for increasingly
precise methods of representing these numbers grew.
Bec
Proofs Involving the Size of 3 Structure
Frequently in Computer Science, we need formulas and theorems to "scale up" to large
problems. For example, when we looked at matrix identities involving matrix multiplication
earlier this semester, we restricted o
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Elementary Counting Problems
1) In how many distinguishable ways can we arrange the letters, FRIEND?
6!
2) In how many distinguishable ways can we arrange the letters, FRIEND, if the I and E
must be adjacent to each other?
3) In how many disti
COMP SCI 241 Discrete Mathematics I
Practice Argument Validation
Spring 2010
p (q r )
r s
(q s )
p
(a b ) p
pt
t
a
p (q r )
qr
rp
p
pr
p q
q s
r s
(p q ) (r s )
rt
t
p
pq
q (r s )
r (t w )
pt
w
Here are six valid arguments.
Prove that each one is valid
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hOWmenK 8 Z! is, A" 9v i i 1. 1f n 2-5. M Z 2:
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7K"L8% + 2 : 2 if; 7 .
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r 53;: V. , y L.\ m ' . . . i >
Z Um i 55.: (Muir 6r 6 DIVISIbIIIty Proofs
This is