Transcript class01
Textbook
Discrete Mathematics and Its Applications,
Rosen
6th Edition
McGraw Hill
2006
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Overview
Much of the basic mathematical machinery useful in
computer science will be presented, with
applications.
Students will learn actively the art of creating realworld proofs in these areas,
preparing them for diverse regions of computer science
such as architecture, algorithms, automata, programming
languages, cryptography, and
increasing their general problem-solving abilities in all areas.
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Problems solved using Discrete Math
How many secure passwords?
Probability of winning Texas Hold’em?
How can I encrypt a message?
Shortest paths between two cities using public
transportation?
How many steps required to sort 10,000 numbers?
Is this algorithm correct?
How to design a circuit that multiply two integers?
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Why study Discrete Maths?
Proof
Ability to understand and create mathematical
argument
Gateway to more advanced CS courses
Data structures, algorithms, automata theory,
formal languages
Database, networks, operating system, security
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Course content
very approximately in temporal order
Ch.
Ch.
Ch.
Ch.
Ch.
Ch.
Ch.
1:
2:
3:
4:
5:
6:
8:
Logic and Proofs
Sets, Functions, Sequences and Sums
Algorithms, the Integers, and Matrices
Induction and Recursion
Counting
Discrete Probability
Relations
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Propositions
A proposition is a statement that can be
either true or false
“Yongdae has an Apple laptop.”
“Yongdae is a professor.”
“3 = 2 + 1”
“3 = 2 + 2”
Not propositions:
“Are you Bob?”
“x = 7”
“I am heavy.”
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Propositional variables
We use propositional variables to refer to
propositions
Usually are lower case letters starting with p (i.e.
p, q, r, s, etc.)
A propositional variable can have one of two
values: true (T) or false (F)
A proposition can be…
A single variable: p
An operation of multiple variables: p(qr)
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Introduction to Logical Operators
About a dozen logical operators
Similar to algebraic operators + * - /
In the following examples,
p = “Today is Friday”
q = “Today is my birthday”
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Logical operators: Not
A “not” operation switches (negates) the
truth value
Symbol: or ~
p
p
p = “Today is not Friday”
T
F
F
T
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Logical operators: And
An “and” operation is true if both operands
are true
Symbol:
It’s like the ‘A’ in And
pq = “Today is Friday and
today is my birthday”
p
q
pq
T
T
T
T
F
F
F
T
F
F
F
F
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Logical operators: Or
An “or” operation is true if either operands
are true
Symbol:
pq = “Today is Friday or
today is my birthday (or
possibly both)”
p
q
pq
T
T
T
T
F
T
F
T
T
F
F
F
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Logical operators: Conditional 1
A conditional means “if p then q”
Symbol:
pq = “If today is
p
q
Friday, then today
T
T
is my birthday”
T
F
p→q=¬pq
the
antecedent
pq
T
F
F
T
T
F
F
T
the
consequence
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Logical operators: Conditional 2
Let p = “I am elected” and q = “I will lower taxes”
I state: p q = “If I
am elected, then I
will lower taxes”
Consider all
possibilities
p
q
pq
T
T
T
T
F
F
F
T
T
F
F
T
Note that if p is false, then
the conditional is true regardless of whether q is
true or false
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Logical operators: Conditional 3
Alternate ways of stating a conditional:
p implies q
If p, q
p only if q
p is sufficient for q
q if p
q whenever p
q is necessary for p
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Logical operators: Conditional 4
Conditional
Inverse
Converse
Contrapositive
p q
p
q
pq
pq
qp
qp
T T
F
F
T
T
T
T
T F
F
T
F
T
T
F
F T
T
F
T
F
F
T
F F
T
T
T
T
T
T
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Logical operators: Bi-conditional 1
A bi-conditional means “p if and only if q”
Symbol:
Alternatively, it means
pq
p
q
“(if p then q) and
T
T
T
(if q then p)”
T
F
F
Note that a bi-conditional
F
T
F
has the opposite truth values
F
F
T
of the exclusive or
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Logical operators: Bi-conditional 2
Let p = “You take this class” and q = “You
get a grade”
pq
p
q
Then pq means
T
T
T
“You take this class if
and only if you get a
T
F
F
grade”
F
T
F
Alternatively, it means “If
F
F
T
you take this class, then
you get a grade and if you get a grade then
you take (took) this class”
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Boolean operators summary
not
not
and
or
xor
conditional
Bi-conditional
p
q
p
q
pq
pq
pq
pq
pq
T
T
F
F
T
T
F
T
T
T
F
F
T
F
T
T
F
F
F
T
T
F
F
T
T
T
F
F
F
T
T
F
F
F
T
T
Learn what they mean, don’t just memorize
the table!
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Precedence of operators
Just as in algebra, operators have
precedence
4+3*2 = 4+(3*2), not (4+3)*2
Precedence order (from highest to lowest):
¬→↔
The first three are the most important
This means that p q ¬r → s ↔ t
yields: (p (q (¬r)) → s) ↔ (t)
Not is always performed before any other
operation
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Translating English Sentences
Question 7 from Rosen, p. 17
p = “It is below freezing”
q = “It is snowing”
It is below freezing and it is snowing
pq
It is below freezing but not snowing
p¬q
It is not below freezing and it is not snowing
¬p¬q
It is either snowing or below freezing (or both)
pq
If it is below freezing, it is also snowing
p→q
It is either below freezing or it is snowing, ((pq)¬(pq))(p→¬q)
but it is not snowing if it is below freezing
That it is below freezing is necessary and
p↔q
sufficient for it to be snowing
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Translation Example 2
Heard on the radio:
A study showed that there was a correlation
between the more children ate dinners with their
families and lower rate of substance abuse by
those children
Announcer conclusions:
If children eat more meals with their family, they will
have lower substance abuse
If they have a higher substance abuse rate, then they
did not eat more meals with their family
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Translation Example 3
“I have neither given nor received help on
this exam”
Let p = “I have given help on this exam”
Let q = “I have received help on this exam”
¬p¬q
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Translation Example 4
You can access the Internet from campus
only if you are a computer science major or
you are not a freshman.
a (c f)
You cannot ride the roller coaster if you are
under 4 feet tall unless you are older than 16
years old.
(f s) r
r ( f s)
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Boolean Searches
(2011 OR 5471) AND yongdae AND “computer science”
Note that Google requires you to capitalize
Boolean operators
Google defaults to AND; many others do not
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Bit Operations
Boolean values can be represented as 1
(true) and 0 (false)
A bit string is a series of Boolean values.
Length of the string is the number of bits.
10110100 is eight Boolean values in one string
We can then do operations on these Boolean
strings
Each column is its own
Boolean operation
01011010
10110100
11101110
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