Transcript Sets

DISCRETE COMPUTATIONAL
STRUCTURES
CSE 2353
Fall 2005
CSE 2353 OUTLINE
1.
2.
3.
4.
5.
6.
7.
8.
Sets
Logic
Proof Techniques
Integers and Induction
Relations and Posets
Functions
Counting Principles
Boolean Algebra
CSE 2353 OUTLINE
1.Sets
2.
3.
4.
5.
6.
7.
8.
Logic
Proof Techniques
Integers and Induction
Relations and Posets
Functions
Counting Principles
Boolean Algebra
Sets: Learning Objectives
Learn about sets
Explore various operations on sets
Become familiar with Venn diagrams
CS:
Learn how to represent sets in computer memory
Learn how to implement set operations in programs
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Sets
Definition: Well-defined collection of distinct
objects
Members or Elements: part of the collection
Roster Method: Description of a set by listing the
elements, enclosed with braces
Examples:
Vowels = {a,e,i,o,u}
Primary colors = {red, blue, yellow}
Membership examples
“a belongs to the set of Vowels” is written as: a 
Vowels
“j does not belong to the set of Vowels: j  Vowels
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Sets
Set-builder method
A = { x | x  S, P(x) } or A = { x  S | P(x) }
 A is the set of all elements x of S, such that x
satisfies the property P
Example:
If X = {2,4,6,8,10}, then in set-builder notation, X
can be described as
X = {n  Z | n is even and 2  n  10}
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Sets
 Standard Symbols which denote sets of numbers
 N : The set of all natural numbers (i.e.,all positive integers)
 Z : The set of all integers
 Z+ : The set of all positive integers
 Z* : The set of all nonzero integers
 E : The set of all even integers
 Q : The set of all rational numbers
 Q* : The set of all nonzero rational numbers
 Q+ : The set of all positive rational numbers
 R : The set of all real numbers
 R* : The set of all nonzero real numbers
 R+ : The set of all positive real numbers
 C : The set of all complex numbers
 C* : The set of all nonzero complex numbers
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Sets
Subsets
“X is a subset of Y” is written as X  Y
“X is not a subset of Y” is written as X
Y
Example:
 X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g}
Y  X, since every element of Y is an element of X
Y
Z, since a  Y, but a  Z
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Sets
Superset
X and Y are sets. If X  Y, then “X is contained in
Y” or “Y contains X” or Y is a superset of X, written
YX
Proper Subset
X and Y are sets. X is a proper subset of Y if X 
Y and there exists at least one element in Y that is
not in X. This is written X  Y.
Example:
 X = {a,e,i,o,u}, Y = {a,e,i,o,u,y}
X  Y , since y  Y, but y  X
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Sets
Set Equality
X and Y are sets. They are said to be equal if every
element of X is an element of Y and every element
of Y is an element of X, i.e. X  Y and Y  X
Examples:
{1,2,3} = {2,3,1}
X = {red, blue, yellow} and Y = {c | c is a primary
color} Therefore, X=Y
Empty (Null) Set
A Set is Empty (Null) if it contains no elements.
The Empty Set is written as 
The Empty Set is a subset of every set
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Sets
Finite and Infinite Sets
X is a set. If there exists a nonnegative integer n
such that X has n elements, then X is called a
finite set with n elements.
If a set is not finite, then it is an infinite set.
Examples:
 Y = {1,2,3} is a finite set
 P = {red, blue, yellow} is a finite set
 E , the set of all even integers, is an infinite set
  , the Empty Set, is a finite set with 0 elements
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Sets
Cardinality of Sets
Let S be a finite set with n distinct elements,
where n ≥ 0. Then |S| = n , where the cardinality
(number of elements) of S is n
Example:
If P = {red, blue, yellow}, then |P| = 3
Singleton
 A set with only one element is a singleton
Example:
H = { 4 }, |H| = 1, H is a singleton
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Sets
Power Set
For any set X ,the power set of X ,written P(X),is
the set of all subsets of X
Example:
If X = {red, blue, yellow}, then P(X) = {  , {red},
{blue}, {yellow}, {red,blue}, {red, yellow}, {blue,
yellow}, {red, blue, yellow} }
Universal Set
An arbitrarily chosen, but fixed set
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Sets
Venn Diagrams
Abstract visualization
of a Universal set, U
as a rectangle, with all
subsets of U shown as
circles.
Shaded portion
represents the
corresponding set
Example:
In Figure 1, Set X,
shaded, is a subset of
the Universal set, U
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Sets
Union of Sets
Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then
X∪Y = {1,2,3,4,5,6,7,8,9}
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Sets
Intersection of Sets
Example:
If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then X ∩ Y = {5}
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Sets
Disjoint Sets
Example:
If X = {1,2,3,4,} and Y = {6,7,8,9}, then X ∩ Y = 
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Sets
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Sets
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Sets
 The union and intersection
of three,four,or even
infinitely many sets can be
considered
 For a finite collection of n
sets, X1, X2, … Xn where n
≥2:
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Sets
Index Set
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Sets
Example:
If A = {a,b,c}, B = {x, y, z} and C = {1,2,3} then
A ∩ B =  and B ∩ C =  and A ∩ C = .
Therefore, A,B,C are pairwise disjoint
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Sets
Difference
• Example:
If X = {a,b,c,d} and Y =
{c,d,e,f}, then X – Y =
{a,b} and Y – X = {e,f}
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Sets
Complement
Example:
If U = {a,b,c,d,e,f} and X = {c,d,e,f}, then X’ = {a,b}
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Sets
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Sets
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Sets
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Sets
Ordered Pair
X and Y are sets. If x  X and y  Y, then an ordered
pair is written (x,y)
Order of elements is important. (x,y) is not necessarily
equal to (y,x)
Cartesian Product
 The Cartesian product of two sets X and Y ,written X × Y
,is the set
 X × Y ={(x,y)|x ∈ X , y ∈ Y}
For any set X, X ×  =  =  × X
 Example:
 X = {a,b}, Y = {c,d}
X × Y = {(a,c), (a,d), (b,c), (b,d)}
Y × X = {(c,a), (d,a), (c,b), (d,b)}
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Computer Representation of Sets
 A Set may be stored in a computer in an array as an
unordered list
Problem: Difficult to perform operations on the set.
Linked List
 Solution: use Bit Strings (Bit Map)
A Bit String is a sequence of 0s and 1s
Length of a Bit String is the number of digits in the
string
Elements appear in order in the bit string
A 0 indicates an element is absent, a 1 indicates that the
element is present
 A set may be implemented as a file
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Computer Implementation of Set Operations
Bit Map
File
Operations
Intersection
Union
Element of
Difference
Complement
Power Set
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Special “Sets” in CS
Multiset
Ordered Set
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CSE 2353 OUTLINE
1. Sets
2.Logic
3. Proof Techniques
4. Relations and Posets
5. Functions
6. Counting Principles
7. Boolean Algebra
Logic: Learning Objectives
 Learn about statements (propositions)
 Learn how to use logical connectives to combine statements
 Explore how to draw conclusions using various argument forms
 Become familiar with quantifiers and predicates
 CS
 Boolean data type
 If statement
 Impact of negations
 Implementation of quantifiers
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Mathematical Logic
Definition: Methods of reasoning, provides rules
and techniques to determine whether an
argument is valid
Theorem: a statement that can be shown to be
true (under certain conditions)
Example: If x is an even integer, then x + 1 is an
odd integer
This statement is true under the condition that x is
an integer is true
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Mathematical Logic
A statement, or a proposition, is a declarative
sentence that is either true or false, but not both
Lowercase letters denote propositions
Examples:
p: 2 is an even number (true)
q: 3 is an odd number (true)
r: A is a consonant (false)
The following are not propositions:
p: My cat is beautiful
q: Are you in charge?
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Mathematical Logic
 Truth value
 One of the values “truth” or “falsity” assigned to a
statement
 True is abbreviated to T or 1
 False is abbreviated to F or 0
 Negation
 The negation of p, written ∼p, is the statement obtained
by negating statement p
Truth values of p and ∼p are opposite
Symbol ~ is called “not” ~p is read as as “not p”
Example:
p: A is a consonant
~p: it is the case that A is not a consonant
q: Are you in charge?
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Mathematical Logic
 Truth Table
 Conjunction
 Let p and q be statements.The conjunction of p and q,
written p ^ q , is the statement formed by joining statements p
and q using the word “and”
The statement p∧q is true if both p and q are true;
otherwise p∧q is false
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Mathematical Logic
Conjunction
Truth Table for
Conjunction:
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Mathematical Logic
Disjunction
Let p and q be statements. The disjunction of p
and q, written p v q , is the statement formed by
joining statements p and q using the word “or”
The statement p v q is true if at least one of the
statements p and q is true; otherwise p v q is
false
The symbol v is read “or”
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Mathematical Logic
Disjunction
Truth Table for
Disjunction:
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Mathematical Logic
Implication
Let p and q be statements.The statement “if p
then q” is called an implication or condition.
The implication “if p then q” is written p  q
p  q is read:
“If p, then q”
“p is sufficient for q”
q if p
q whenever p
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Mathematical Logic
Implication
Truth Table for Implication:
p is called the hypothesis, q is called the
conclusion
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Mathematical Logic
Implication
Let p: Today is Sunday and q: I will wash the car.
The conjunction p  q is the statement:
p  q : If today is Sunday, then I will wash the car
The converse of this implication is written q  p
If I wash the car, then today is Sunday
The inverse of this implication is ~p  ~q
If today is not Sunday, then I will not wash the car
The contrapositive of this implication is ~q  ~p
If I do not wash the car, then today is not Sunday
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Mathematical Logic
Biimplication
Let p and q be statements. The statement “p if
and only if q” is called the biimplication or
biconditional of p and q
The biconditional “p if and only if q” is written p 
q
p  q is read:
“p if and only if q”
“p is necessary and sufficient for q”
“q if and only if p”
“q when and only when p”
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Mathematical Logic
Biconditional
Truth Table for the Biconditional:
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Mathematical Logic
 Statement Formulas
 Definitions
 Symbols p ,q ,r ,...,called statement variables
 Symbols ~, , v, →,and ↔ are called logical
^
connectives
1) A statement variable is a statement formula
2) If A and B are statement formulas, then the
expressions (~A ), (A B) , (A v B ), (A → B )
^
and (A ↔ B ) are statement formulas
 Expressions are statement formulas that are
constructed only by using 1) and 2) above
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Mathematical Logic
Precedence of logical connectives is:
~ highest

^
second highest
 v third highest
→ fourth highest
↔ fifth highest
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Mathematical Logic
Example:
Let A be the statement formula (~(p v q )) → (q
Truth Table for A is:
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^p)
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Mathematical Logic
Tautology
A statement formula A is said to be a tautology if
the truth value of A is T for any assignment of the
truth values T and F to the statement variables
occurring in A
Contradiction
A statement formula A is said to be a
contradiction if the truth value of A is F for any
assignment of the truth values T and F to the
statement variables occurring in A
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Mathematical Logic
Logically Implies
A statement formula A is said to logically imply a
statement formula B if the statement formula A →
B is a tautology. If A logically implies B, then
symbolically we write A → B
Logically Equivalent
A statement formula A is said to be logically
equivalent to a statement formula B if the
statement formula A ↔ B is a tautology. If A is
logically equivalent to B , then symbolically we
write A ≡ B
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Mathematical Logic
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Mathematical Logic
 Proof of (~p
^q)→
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(~(q →p ))
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Mathematical Logic
Proof of (~p
^
q ) → (~(q →p )) [continued]
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Validity of Arguments
Proof: an argument or a proof of a theorem
consists of a finite sequence of statements
ending in a conclusion A1 , A2 , A3 , ..., An1 , An
Argument: a finite sequence
of statements.
The final statement, An , is the conclusion, and
the statements
A1 , A2 , A3 , ..., An1 are the
premises of the argument.
An argument is logically valid if the statement
formula A1 , A2 , A3 , ..., An1  An
is a tautology.
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Validity of Arguments
Valid Argument Forms
Modus Ponens (Method of Affirming)
Modus Tollens (Method of Denying)
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Validity of Arguments
Valid Argument Forms
Disjunctive Syllogisms
Disjunctive Syllogisms
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Validity of Arguments
 Valid Argument Forms
 Hypothetical Syllogism
 Dilemma
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Validity of Arguments
Valid Argument Forms
Conjunctive Simplification
Conjunctive Simplification
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Validity of Arguments
Valid Argument Forms
Disjunctive Addition
Disjunctive Addition
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Validity of Arguments
Valid Argument Forms
Conjunctive Addition
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Quantifiers and First Order Logic
Predicate or Propositional Function
Let x be a variable and D be a set; P(x) is a
sentence
Then P(x) is called a predicate or propositional
function with respect to the set D if for each
value of x in D, P(x) is a statement; i.e., P(x) is
true or false
Moreover, D is called the domain of the
discourse and x is called the free variable
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Quantifiers and First Order Logic
Predicate or Propositional Function
Example:
 Q(x,y) : x > y, where the Domain is the set of
integers
 Q is a 2-place predicate
 Q is T for Q(4,3) and Q is F for Q (3,4)
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Quantifiers and First Order Logic
Universal Quantifier
Let P(x) be a predicate and let D be the domain
of the discourse. The universal quantification of
P(x) is the statement:
For all x, P(x)
or
For every x, P(x)
The symbol  is read as “for all and every”
 x P ( x)
 Two-place predicate: xy P( x, y )
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Quantifiers and First Order Logic
Existential Quantifier
Let P(x) be a predicate and let D be the domain
of the discourse. The existential quantification of
P(x) is the statement:
There exists x, P(x)
The symbol  is read as “there exists”
 x P ( x )
 Bound Variable
The variable appearing in:
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x P ( x)
or
x P ( x )
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Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s Laws)

~ x P( x)  x ~ P( x)
Example:
 If P(x) is the statement “x has won a race” where
the domain of discourse is all runners, then the
universal quantification of P(x) is x P ( x) , i.e.,
every runner has won a race. The negation of this
statement is “it is not the case that every runner
has won a race. Therefore there exists at least one
runner who has not won a race. Therefore: x ~ P ( x)
and so,
~ x P( x)  x ~ P( x)
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Quantifiers and First Order Logic
Negation of Predicates (DeMorgan’s
Laws)
 ~ x P( x)  x ~ P( x)
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Logic and CS
Logic is basis of ALU
Logic is crucial to IF statements
AND
OR
NOT
Implementation of quantifiers
Looping
Database Query Languages
Relational Algebra
Relational Calculus
SQL
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Inductions
5. Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Proof Technique: Learning Objectives
 Learn various proof techniques
 Direct
 Indirect
 Contradiction
 Induction
 Practice writing proofs
CS: Why study proof techniques?
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Proof Techniques
Theorem
Statement that can be shown to be true (under
certain conditions)
Typically Stated in one of three ways
As Facts
As Implications
As Biimplications
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Proof Techniques
Direct Proof or Proof by Direct Method
Proof of those theorems that can be expressed in
the form ∀x (P(x) → Q(x)), D is the domain of
discourse
Select a particular, but arbitrarily chosen, member
a of the domain D
Show that the statement P(a) → Q(a) is true.
(Assume that P(a) is true
Show that Q(a) is true
By the rule of Universal Generalization (UG),
∀x (P(x) → Q(x)) is true
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Proof Techniques
Indirect Proof
The implication p → q is equivalent to the
implication (∼q → ∼p)
Therefore, in order to show that p → q is true,
one can also show that the implication
(∼q → ∼p) is true
To show that (∼q → ∼p) is true, assume that the
negation of q is true and prove that the negation
of p is true
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Proof Techniques
Proof by Contradiction
Assume that the conclusion is not true and then
arrive at a contradiction
Example: Prove that there are infinitely many prime
numbers
Proof:
Assume there are not infinitely many prime numbers,
therefore they are listable, i.e. p1,p2,…,pn
Consider the number q = p1p2…pn+1. q is not
divisible by any of the listed primes
Therefore, q is a prime. However, it was not listed.
Contradiction! Therefore, there are infinitely many
primes
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Proof Techniques
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Proof Techniques
Proof of Biimplications
To prove a theorem of the form ∀x (P(x) ↔
Q(x )), where D is the domain of the
discourse, consider an arbitrary but fixed
element a from D. For this a, prove that the
biimplication P(a) ↔ Q(a) is true
The biimplication p ↔ q is equivalent to
(p → q) ∧ (q → p)
Prove that the implications p → q and q → p
are true
Assume that p is true and show that q is true
Assume that q is true and show that p is true
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Proof Techniques
Proof of Equivalent Statements
Consider the theorem that says that statements
p,q and r are equivalent
Show that p → q, q → r and r → p
Assume p and prove q. Then assume q and prove
r Finally, assume r and prove p
Or, prove that p if and only if q, and then q if and
only if r
Other methods are possible
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Other Proof Techniques
Vacuous
Trivial
Contrapositive
Counter Example
Divide into Cases
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Proof Basics
You can not prove by
example
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Proofs in Computer Science
Proof of program correctness
Proofs are used to verify approaches
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CSE 2353 OUTLINE
1. Sets
2. Logic
3. Proof Techniques
4. Integers and Induction
5. Relations and Posets
6. Functions
7. Counting Principles
8. Boolean Algebra
Learning Objectives
Learn about the basic properties of integers
Explore how addition and subtraction operations
are performed on binary numbers
Learn how the principle of mathematical
induction is used to solve problems
CS
Become aware how integers are represented in
computer memory
Looping
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Integers
Properties of Integers
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Integers
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Integers
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Integers
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Integers
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Integers
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Integers
The div and mod operators
div
 a div b = the quotient of a and b obtained by dividing a
on b.
Examples:
8 div 5 = 1
13 div 3 = 4
mod
a mod b = the remainder of a and b obtained by dividing
a on b
8 mod 5 = 3
13 mod 3 = 1
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Integers
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Integers
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Integers
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Integers
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Integers
 Relatively Prime
Number
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Integers
 Least Common Multiples
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Representation of Integers in Computer
Electrical signals are used inside the
computer to process information
Two types of signals
Analog
Continuous wave forms used to represent such things as
sound
Examples: audio tapes, older television signals, etc.
Digital
Represent information with a sequence of 0s and 1s
Examples: compact discs, newer digital HDTV signals
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Representation of Integers in Computers
Digital Signals
0s and 1s – 0s represent low voltage, 1s high
voltage
Digital signals are more reliable carriers of
information than analog signals
Can be copied from one device to another
with exact precision
Machine language is a sequence of 0s and 1s
The digit 0 or 1 is called a binary digit , or bit
A sequence of 0s and 1s is sometimes referred to
as binary code
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Representation of Integers in Computers
Decimal System or Base-10
The digits that are used to represent numbers in base
10 are 0,1,2,3,4,5,6,7,8, and 9
Binary System or Base-2
Computer memory stores numbers in machine
language, i.e., as a sequence of 0s and 1s
Octal System or Base-8
Digits that are used to represent numbers in base 8
are 0,1,2,3,4,5,6, and 7
Hexadecimal System or Base-16
Digits and letters that are used to represent numbers
in base 16 are 0,1,2,3,4,5,6,7,8,9,A ,B ,C ,D ,E , and F
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Representation of Integers in Computers
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Representation of Integers in Computers
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Representation of Integers in Computers
Two’s Complements and Operations on
Binary Numbers
In computer memory, integers are
represented as binary numbers in fixed-length
bit strings, such as 8, 16, 32 and 64
Assume that integers are represented as 8-bit
fixed-length strings
Sign bit is the MSB (Most Significant Bit)
Leftmost bit (MSB) = 0, number is positive
Leftmost bit (MSB) = 1, number is negative
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Representation of Integers in Computers
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Representation of Integers in Computers
One’s Complements and Operations on Binary
Numbers
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Representation of Integers in Computers
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Representation of Integers in Computers
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Representation of Integers in Computers
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Representation of Integers in Computers
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Representation of Integers in Computers
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Mathematical Deduction
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Mathematical Deduction
 Proof of a mathematical statement by the principle of
mathematical induction consists of three steps:
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Mathematical Deduction
Assume that when a domino is knocked over, the
next domino is knocked over by it
Show that if the first domino is knocked over,
then all the dominoes will be knocked over
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Mathematical Deduction
Let P(n) denote the statement that then nth
domino is knocked over
Show that P(1) is true
Assume some P(k) is true, i.e. the kth domino is
knocked over for some
k 1
Prove that P(k+1) is true, i.e.
P( k )  P ( k  1)
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Mathematical Deduction
Assume that when a staircase is climbed, the
next staircase is also climbed
Show that if the first staircase is climbed then
all staircases can be climbed
Let P(n) denote the statement that then nth
staircase is climbed
It is given that the first staircase is climbed, so
P(1) is true
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Mathematical Deduction
Suppose some P(k) is true, i.e. the kth
staircase is climbed for some
k 1
By the assumption, because the kth staircase
was climbed, the k+1st staircase was
climbed
Therefore, P(k) is true, so
P ( k )  P ( k  1)
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Mathematical Deduction
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Mathematical Deduction
We can associate a predicate, P(n). The
predicate P(n) is such that:
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Prime Numbers
For any positive integer n > 1, the integers 1 and
n are called the trivial positive divisors of n
An integer n > 1 is a prime integer if and only if n
has only trivial positive divisors
An integer n > 1 is a composite integer if and
only if n has a nontrivial positive divisor
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Prime Numbers
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Prime Numbers
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Prime Numbers
Example:
Consider the integer 131. Observe that 2 does not divide 131. We now find all
odd primes p such that p2  131. These primes are 3, 5, 7, and 11. Now none of
3, 5, 7, and 11 divides 131. Hence, 131 is a prime.
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Prime Numbers
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Prime Numbers
Factoring a Positive Integer
The standard factorization of n
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Prime Numbers
 Fermat’s Factoring Method
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Prime Numbers
 Fermat’s Factoring Method
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