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Chapter 5 - Set Theory

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Title: Chapter 5 - Set Theory


1
Chapter 5 - Set Theory
5.1.1
  • 1. Basic Definitions
  • 2. Empty Set, Partitions, Power Set
  • 3. Properties of Sets

2
Section 1. Basic Definitions
5.1.2
  • A Set is a collection of items, called elements.
  • 1, 2, 3
  • x Î R x2 gt 5
  • S Tom, Sue, Jim

3
5.1.3
  • We use ellipses to simplify things
  • 1, 2, 3, , 10
  • 1, 2, 3,
  • , -2, -1, 0, 1, 2,
  • Be careful! (1, 2, ???)

4
5.1.4
  • We relate an item in the set with the set using
    the Î (element of) relation.
  • x Î x, y, z
  • 1, 2 Î 1, 2, 1, 2, 3.

5
Special Sets
5.1.5
  • We refer to specific sets of numbers so often
    that we give them special names.
  • These sets, and their corresponding symbols, will
    be referenced throughout this course.

6
Natural Numbers
5.1.6
  • We define the Natural Numbers to beN 0, 1,
    2, 3,
  • Note that the Naturals are closed under
    addition and multiplication.

7
The Integers
5.1.7
  • We define the Integers to beZ , -2, -1, 0,
    1, 2, 3,
  • Note that Z is closed under addition,
    subtraction, and multiplication.

8
The Rational Numbers
5.1.8
  • We define the Rationals to beQ p/q p,q Î
    Z and q ¹ 0
  • Note that Q is closed under addition,
    subtraction, multiplication, and non-zero
    division.

9
The Rational Numbers
5.1.9
  • Alternatively, we can view Q as the set of all
    infinite, repeating decimal expansions.
  • 7.35 7.3500000 Î Q
  • 1.234234234 Î Q
  • However, p Ï Q

10
The Irrational Numbers
5.1.10
  • I all infinite, nonrepeating decimals
  • Obviously, irrational numbers are impossible to
    write down exactly.
  • We use symbols to represent special values such
    as p, e, and Ö2.
  • The Irrationals are not closed under or .

11
The Real Numbers
5.1.11
  • R all decimal expansions
  • The Real Numbers are created by adjoining the
    Rationals with the Irrationals.
  • The Reals are closed under all operations and
    satisfy the Field Axioms (see Appendix A, p.
    695).
  • The Reals form a continuum we use the Real
    Number Line to represent this.

12
The Complex Numbers
5.1.12
  • The Reals fall short when solving simple
    polynomial equations like x2 1 0.
  • The Complex Numbers patch this hole.
  • C a bi a,b Î R and i Ö (-1)
  • Use the Complex Plane to represent these numbers.
  • The Complex Numbers are also a field.

13
Subsets
5.1.13
  • If A and B are sets, A is called a subset of B,
    denoted A Í B, provided every element of A is an
    element of B.
  • So, A Í B means "x, if x Î A, then x Î B.
  • We also say, A is contained in B or B
    contains A to show this relationship.
  • Equivalently, we denote A Ë B provided x ' x Î
    A and x Ï B.

14
Examples of Subsets
5.1.14
  • If A 1, 2, 3 and B 0, 1, 2, 3, 4, then
    clearly A Í B.
  • 1, 2 Í 1, 2, 1,2.
  • Q Í R and Z Í Q and N Í Z.
  • a, b, c is a proper subset of a, b, c, d.
  • a, b, c is an improper subset of a, b, c.
  • We denote interval subsets of R asa, b) x Î
    R a x lt b. So 2, 5) Í 0,5.

15
Set Equality
5.1.15
  • We say sets A and B are equal (A B) if every
    element of A is in B and every element of B is in
    A.
  • Thus, A B means A Í B and B Í A.
  • For example 1, 2, 3 1, 2, 3, butA 1, 2,
    3 ¹ 1, 2, 3, 4 B, since 4 Î B but 4 Ï A.
  • Also, a, b) ¹ a, b since b is only in a, b.

16
Operations on Sets
5.1.16
  • Given sets A and B, which are subsets of a
    universal set, U, we define the following
  • (Union) A È B x Î U x Î A or x Î B.
  • (Intersection) A Ç B x Î U x Î A and x Î B.
  • (Difference or Relative Complement) A - B x Î
    U x Î A and x Ï B.
  • (Complement) Ac x Î U x Ï A.
  • Note that Ac U - A.

17
Examples of Set Operations
5.1.17
  • Let U R, A 1, 3 and B (2, 4).
  • A È B 1, 4)
  • A Ç B (2, 3
  • A - B 1, 2
  • B - A (3, 4)
  • Ac (-, 1) È (3, )
  • Bc (-, 2 È 4, )

18
Cartesian Products
5.1.18
  • Given two sets, A and B, we define the Cartesian
    Product, A B (a, b) a Î A and b Î B.
  • The element (a, b) is called an ordered pair,
    since (a, b) and (b, a) are distinct if a ¹ b.
  • If A 1, 2, 3 and B 8, 9, thenA B
    (1, 8), (1, 9), (2, 8), (2, 9), (3, 8), (3,
    9)B A (8, 1), (8, 2), (8, 3), (9, 1), (9,
    2), (9, 3)

19
Generalized Cartesian Products
5.1.19
  • Given three sets, A, B, and C, we define their
    Cartesian Product byA B C (a, b, c) a Î
    A, b Î B and c Î C.
  • Although similar, we note that A B C and (A
    B) C are not, technically, the same since one
    contains (a, b, c) and the other ((a, b), c).
  • In general, we define A1A2A3...An to
    be(a1,a2,a3,,an) a1ÎA1,a2ÎA2,a3ÎA3,, anÎAn

20
Formal Languages
5.1.20
  • Let S be a finite set, which we will, henceforth,
    call an alphabet.
  • A string of characters of the alphabet S (or a
    string over S ) is either (1) an ordered n-tuple
    of elements of S written without parentheses or
    commas, or (2) the null string e, which has no
    characters.

21
Formal Languages (contd.)
5.1.21
  • If S 1, 2, 3, then 131221 is a string of
    length 6 over S.
  • 131221 (1, 3, 1, 2, 2, 1) Î SSSSSS.
  • Clearly, the length of a string, s, over an
    alphabet S, is the number of characters of S that
    are in s.
  • We denote the function L(s) to be this length.
  • Hence, if S 0, 1, then L(101100111) 9.

22
Formal Languages (contd.)
5.1.22
  • Any set of strings over an alphabet is called a
    formal language over the alphabet.
  • Let S be an alphabet and n Î N
  • Sn strings over S with L(s) n
  • Gn strings over S with L(s) n
  • S strings over S of finite length.

23
Examples
5.1.23
  • Let S 0, 1
  • S3 000, 001, 010, 011, 100, 101, 110, 111
  • G3 e, 0, 1, 00, 01, 10, 11, 000, 001, 010,
    011, 100, 101, 110, 111
  • S e, 0, 1, 00, 01, 10, 11, 000, 001, 010,
    011, 100, 101, 110, 111, , 000000, , 111111,
    ....

24
Section 5.3
5.3.24
  • The Empty Set
  • Partitions
  • Power Sets
  • Boolean Algebras

25
The Empty Set
5.3.25
  • The unique set containing no elements is called
    the empty set, denoted Æ or .
  • Theorem If A is a set, then Æ Í A.
  • Corollary Æ is unique.
  • Why? Since Æ1 Í Æ2 and Æ2 Í Æ1 we conclude Æ1
    Æ2 .

26
Set Operations With Æ
5.3.26
  • For any set A from a universal set U
  • A Ç Æ Æ
  • A È Æ A
  • A Ç Ac Æ
  • A È Ac U
  • Uc Æ and Æc U.

27
Partitions of a Set
5.3.27
  • Two sets are called disjoint if they have no
    elements in common. That is, A and B are
    disjoint provided A Ç B Æ.
  • Theorem If A and B are any sets, then (A - B)
    and B are disjoint.
  • A collection of sets A1,A2,,An is called
    mutually or pairwise disjoint if Ai Ç Aj Æ
    whenever i ¹ j.

28
Partitions of a Set (contd.)
5.3.28
  • A collection of sets A1,A2,,An is called a
    partition of a set A provided1. A1,A2,,An
    is mutually disjoint2. A1 È A2 È ... È An A.
  • 1, 2, 3,4, 5,6, 7, 8 partitions?
  • 1,2,3,...,-1,-2,-3,...,0 partitions?
  • Q, I partitions?

29
Power Sets
5.3.29
  • If A is a set, the Power Set of A, denoted P (A),
    is the set of all subsets of A.
  • Since Æ Í A, we conclude Æ Î P (A), and A Í A
    implies A Î P (A).
  • If A 0,1, P (A) Æ,0,1,0,1
  • Theorem If A and B are sets with A Í B, then P
    (A) Í P (B).
  • Theorem If A n, then P (A) 2n.

30
Boolean Algebras
5.3.30
  • If A is a set, the collection A, , is called
    a Boolean Algebra if
  • 1. " a,bÎA, a b b a and a b b a
  • 2. " a,b,cÎA, (a b) c a (b c) and (a
    b) c a (b c)
  • 3. " a,b,cÎA, a (b c) (a b) (a c)
    and a (b c) (a b) (a c)
  • 4. ! 0,1ÎA ' " aÎA, a 0 a and a 1 a
  • 5. " aÎA, bÎA ' a b 1 and a b 0

31
Chapter 1. Symbolic Logic
1.1.31
  • Logical Form and Equivalence
  • Conditional Statements
  • Valid and Invalid Arguments
  • Digital Logic Circuits (Boolean Polynomials)

32
Logic of Compound Statements
1.1.32
  • A statement (or proposition) is a sentence that
    is true (T) or false (F), but not both or
    neither.
  • Examples
  • Today is Monday.
  • x is even and x gt 7.
  • If x2 4, then x 2 or x -2.

33
Counterexamples
1.1.33
  • If a sentence cannot be judged to be T or F or is
    not even a sentence, it cannot be a statement.
  • Examples
  • Open the door! (imperative)
  • Did you open the door? (interrogative)
  • If x2 4. (fragment)

34
Compound Statements
1.1.34
  • Denote statements using the symbols p, q, r, ...
  • Denote the operations Ù, Ú, , (to be defined
    shortly), where
  • p Ù q - conjunction of p and q (p and q)
  • p Ú q - disjunction of p and q (p or q)
  • p - negation of p (not p)
  • p q - implication of p and q (p implies q)

35
Compound Statements (contd.)
1.1.35
  • A Compound statement (or statement form) is a
    statement which includes at least one operation
    and one other atomic statement.
  • For example, x 7 and y 2 is a compound
    statement based on the atomic statements p
    x 7 and q y 2.
  • In this instance, we can symbolize the compound
    statement as r p Ù q.

36
Compound Statements (contd.)
1.1.36
  • The Truth Table of a compound statement is the
    collection of all the output truth values
    corresponding to all possible combinations of
    input truth values of the atomic statements.
  • Since each atomic statement can take on 1 of 2
    values, 2 inputs have 4 combinations, 3 inputs
    have 8, 4 inputs have 16, 5 inputs have 32, etc.

37
Logical Operations
1.1.37
  • Negation p p T F F T
  • Conjunction Disjunction p q (p Ù q)
    p q (p Ú q) T T T T T
    T T F F T F T F T
    F F T T F F F F F F

38
Example (p Ú q) Ù r
1.1.38
  • Proceed from left to right p q r (p Ú q)
    r (p Ú q) Ù r
  • T T T T F F
  • T T F T T T
  • T F T T F F
  • T F F T T T
  • F T T T F F
  • F T F T T T
  • F F T F F F
  • F F F F T F

39
Logical Equivalence
1.1.39
  • Two compound statements are logically equivalent
    if they have the same truth table. We denote this
    as p º q.
  • p p (p)
  • T F T
  • F T F hence p º (p).
  • (p Ù q) º p Ù q ?
  • No, since (T Ù F) º T, but (T Ù F) º F.

40
Tautology Contradiction
1.1.40
  • A statement whose truth table is all T is
    called a tautology, denoted as p º t.
  • A statement whose truth table is all F is
    called a contradiction, denoted as p º c.
  • Clearly, t º c and c º t.
  • Are all logical statements either tautology or
    contradiction?

41
Algebra of Symbolic Logic
1.1.41
  • Commutative Laws
  • p Ù q º q Ù p
  • p Ú q º q Ú p
  • Associative Laws
  • (p Ù q) Ù r º p Ù (q Ù r)
  • (p Ú q) Ú r º p Ú (q Ú r)
  • Distributive Laws
  • p Ù (q Ú r) º (p Ù q) Ú (p Ù r)
  • p Ú (q Ù r) º (p Ú q) Ù (p Ú r)

42
Algebra of Symbolic Logic
1.1.42
  • Identity Laws
  • p Ù t º p
  • p Ú c º p
  • Negation Laws
  • p Ù p º c
  • p Ú p º t
  • Double Negative Laws (p) º p
  • Negations of t and c t º c c º t

43
Algebra of Symbolic Logic
1.1.43
  • Idempotent Laws p Ù p º p p Ú p º p
  • DeMorgans Laws
  • (p Ù q) º p Ú q
  • (p Ú q) º p Ù q
  • Universal Bound Laws p Ù c º c p Ú t º t
  • Absorption Laws
  • p Ù (p Ú q) º p
  • p Ú (p Ù q) º p

44
Section 1.2
1.2.44
  • Conditional Statements
  • Logical Equivalences Involving Conditionals
  • Converses, Inverses, and Contrapositives
  • Biconditional Statements

45
Conditional Statements
1.2.45
  • If p and q are statement variables, the
    conditional or implication of q by p is If p
    then q or p implies q and is denoted by p
    q.
  • The truth table of the implication operator is
  • p q p q
  • T T T
  • T F F
  • F T T
  • F F T
  • Example If you mow my lawn, Ill pay 20.

46
Hypotheses Conclusions
1.2.46
  • In the form If p then q the statement p is
    called the hypothesis and the statement q is the
    conclusion.
  • Conditionals form the basis of deductive
    reasoning. (Aristotilean Logic)
  • In looking at the truth table, we consider the
    cases where the hypothesis is false to yield
    vacuous results. The interesting cases are when
    the hypothesis is true.

47
Logical Equivalences
1.2.47
  • In the framework of symbolic logic, the
    implication operator would seem to be a new and
    distinct process.
  • However, this is not the case!
  • Theorem p q º p Ú q.
  • Thus, we can always rewrite an implication as a
    disjunction.
  • Corollary (p q) º p Ù q.

48
Negation of a Conditional
1.2.48
  • From the previous corollary, the negation of p
    q is p Ù q.
  • For example, the negation of
  • If today is Sunday, then I wash my car.
  • is
  • Today is Sunday and I do not wash my car.

49
Converse of a Conditional
1.2.49
  • Given the statement p q, we define its converse
    to be the statement q p.
  • For example, the converse of
  • If today is Sunday, then I wash my car.
  • is
  • If I wash my car, then today is Sunday.

50
Contrapositive of a Conditional
1.2.50
  • Given the statement p q, we define its
    contrapositive to be the statement q p.
  • For example, the contrapositive of
  • If today is Sunday, then I wash my car.
  • is
  • If I do not wash my car, then today is not Sunday.

51
Inverse of a Conditional
1.2.51
  • Given the statement p q, we define its inverse
    to be the statement p q.
  • For example, the inverse of
  • If today is Sunday, then I wash my car.
  • is
  • If today is not Sunday, then I do not wash my car.

52
Equivalent Forms
1.2.52
  • Theorem Given the statement p q, we have that
    p q º q p.
  • Corollary Given the statement p q, we have
    that q p º p q.
  • Therefore from the above, we see that a
    conditional and its contrapositive are logically
    equivalent.
  • Moreover, the statements converse and inverse
    forms are logically equivalent to each other.

53
Biconditional Statements
1.2.53
  • Definition Given the statement variables p and
    q, the biconditional of p and q is read, p if
    and only if q, denoted p q and means that both
    p q and q p .
  • By direct calculation p q p q
  • T T T
  • T F F
  • F T F
  • F F T

54
Using the Biconditional
1.2.54
  • Looking closely at the truth table, we see that
    p q is T whenever p and q have the same truth
    value.
  • Theorem p q is a tautology implies p º q and
    conversely, p º q implies p q is a tautology.
  • This gives us a systematic way to calculate
    logical equivalence, rather then just scan the
    matches of truth values by eye.

55
Section 1.3
1.3.55
  • Valid and invalid argument forms.
  • Special valid argument forms.
  • Dilemmas
  • Fallacies.
  • Contradictions and valid arguments.

56
Valid and Invalid Arguments
1.3.56
  • An argument (or argument form) is a sequence of
    statements.
  • All statements but the final one are called
    premises, assumptions, or hypotheses.
  • The final statement is called the conclusion.
  • An argument form is valid provided its conclusion
    is always true whenever all of its premises are
    true.

57
Valid and Invalid Arguments
1.3.57
  • An argument (or argument form) is a sequence of
    statements.
  • All statements but the final one are called
    premises, assumptions, or hypotheses.
  • The final statement is called the conclusion.
  • An argument form is valid provided its conclusion
    is always true whenever all of its premises are
    true.
  • The truth of the conclusion follows inescapably
    from the truth of the hypotheses.

58
Testing for Validity
1.3.58
  • Identify the premises and conclusion.
  • Construct a truth table for the premises and
    conclusion.
  • Find the critical rows, where all premises are T.
  • For each critical row, if the conclusion is also
    T, then the argument is valid.
  • If at least one critical row leads to a
    conclusion being F, the argument is invalid.
  • If there are no critical rows, the argument is
    vacuously valid.

59
A Valid Argument
1.3.59
  • p Ú (q Ú r)
  • r
  • \ (p Ú q)
  • Truth Table p q r p Ú (q Ú r) r (p Ú
    q)
  • T T T T F T
  • T T F T T T
  • T F T T F T
  • T F F T T T
  • F T T T F T
  • F T F T T T
  • F F T T F F
  • F F F F T F

60
An Invalid Argument
1.3.60
  • p Ú (q Ú r)
  • r
  • \ (p Ú r)
  • Truth Table p q r p Ú (q Ú r) r (p Ú
    r)
  • T T T T F T
  • T T F T T T
  • T F T T F T
  • T F F T T T
  • F T T T F T
  • F T F T T F
  • F F T T F T
  • F F F F T F

61
Special Argument Forms
1.3.61
  • Modus Ponens p q
  • p
  • \ q
  • Truth Table p q p q p q
  • T T T T T
  • T F F T F
  • F T T F T
  • F F T F F
  • Premises If today is Sunday, then I was my car.
  • Today is Sunday.
  • Conclusion I wash my car.

62
Modus Tollens
1.3.62
  • Modus Tollens p q
  • q
  • \ p
  • Truth Table p q p q q p
  • T T T F F
  • T F F T F
  • F T T F T
  • F F T T T
  • Premises If today is Sunday, then I was my car.
  • I do not wash my car.
  • Conclusion Today is not Sunday.

63
Disjunctive Addition
1.3.63
  • Disjunctive Addition p
  • \ p Ú q
  • Truth Table p q p Ú q
  • T T T
  • T F T
  • F T T
  • F F F
  • Premise Today is Sunday.
  • Conclusion Today is Sunday or I wash my car.

64
Conjunctive Simplification
1.3.64
  • Conjunctive Simplification p Ù q
  • \ p
  • also \ q
  • Truth Table p q p Ù q
  • T T T
  • T F F
  • F T F
  • F F F
  • Premise Today is Sunday and I wash my car.
  • Conclusion 1 Today is Sunday.
  • Conclusion 2 I wash my car.

65
Disjunctive Syllogism
1.3.65
  • Disjunctive Syllogism p Ú q p Ú q
  • p q
  • \ q \ p
  • Truth Table p q p Ú q p
  • T T T F
  • T F T F
  • F T T T
  • F F F T
  • Premises Today is Sunday or Saturday.
  • Today is not Sunday.
  • Conclusion Today is Saturday.

66
Hypothetical Syllogism
1.3.66
  • Hypothetical Syllogism p q
  • q r
  • \ p r
  • Premises If x is an integer, then x is a
    rational.
  • If x is a rational, then x is a real.
  • Conclusion If x is an integer, then x is real.

67
Dilemma Division Into Cases
1.3.67
  • Dilemma p Ú q
  • p r
  • q r
  • \ r
  • Premises x is positive or x is negative.
  • If x is positive , then x2 is positive.
  • If x is negative, then x2 is positive.
  • Conclusion x2 is positive.

68
Application Find My Glasses
1.3.68
  • 1. If my glasses are on the kitchen table, then I
    saw them at breakfast.
  • 2. I was reading in the kitchen or I was reading
    in the living room.
  • 3. If I was reading in the living room, then my
    glasses are on the coffee table.
  • 4. I did not see my glasses at breakfast.
  • 5. If I was reading in bed, then my glasses are
    on the bed table.
  • 6. If I was reading in the kitchen, then my
    glasses are on the kitchen table.

69
Find My Glasses (contd.)
1.3.69
  • Let p My glasses are on the kitchen table.
  • q I saw my glasses at breakfast.
  • r I was reading in the living room.
  • s I was reading in the kitchen.
  • t My glasses are on the coffee table.
  • u I was reading in bed.
  • v My glasses are on the bed table.

70
Find My Glasses (contd.)
1.3.70
  • Then the original statements become
  • 1. p q 2. r Ú s 3. r t
  • 4. q 5. u v 6. s p
  • and we can deduce (why?)
  • 1. p q 2. s p 3. r Ú s 4. r t
  • q p s r
  • \ p \ s \ r \ t
  • Hence the glasses are on the coffee table!

71
Fallacies
1.3.71
  • A fallacy is an error in reasoning that results
    in an invalid argument.
  • Three common fallacies
  • Using vague or ambiguous premises
  • Begging the question
  • Jumping to a conclusion.
  • Two dangerous fallacies
  • Converse error
  • Inverse error.

72
Converse Error
1.3.72
  • If Zeke cheats, then he sits in the back row.
  • Zeke sits in the back row.
  • \ Zeke cheats.
  • The fallacy here is caused by replacing the
    impication (Zeke cheats sits in back) with its
    biconditional form (Zeke cheats sits in back),
    implying the converse (sits in back Zeke
    cheats).

73
Inverse Error
1.3.73
  • If Zeke cheats, then he sits in the back row.
  • Zeke does not cheat.
  • \ Zeke does not sit in the back row.
  • The fallacy here is caused by replacing the
    impication (Zeke cheats sits in back) with its
    inverse form (Zeke does not cheat does not sit
    in back), instead of the contrapositive (does not
    sit in back Zeke does not cheat).

74
Contradiction Rule
1.3.74
  • If you can show that assuming statement p is
    false leads logically to a contradiction, then
    you can conclude that p is true.
  • In argument form p c \ p
  • This is the logical heart of the proof method
    called Proof by Contradiction.

75
Section 1.4
1.4.75
  • Digital Logic Circuits
  • Boolean Polynomials
  • Normal Forms (Disjunctive/Conjunctive)
  • Designing Circuits with Specified Conditions
  • Showing Two Circuits Are Equivalent

76
Digital Logic Circuits
1.4.76
  • Developed by Claude Shannon in 1938 to model
    telephone switching circuits

x AND y
x
y
Series Switch
x
x OR y
y
Parallel Switch
77
Logical Gates
1.4.77
  • Instead of working with switches, we model
    digital circuits using gates AND-gates,
    OR-gates, and NOT-gates.
  • We draw these as

OR
x
x
NOT
78
Notation
1.4.78
  • Modeling digital circuits leads to the equivalent
    analysis of symbolic logic.
  • Symbolic Logic Digital Circuits
  • T, t 1, 1
  • F, c 0, 0
  • p, q, r, ... x, y, z, ...
  • p x
  • p Ù q xy
  • p Ú q x y

79
Boolean Polynomials
1.4.79
  • When modeling, we use Boolean polynomials to
    describe algebraically the function of a
    combinatorial circuit.
  • A combinatorial circuit is one in which the
    output at any time depends on the inputs at the
    previous time. (i.e. no feedback loops)
  • A Boolean polynomial is a function which takes
    0,1 inputs and outputs a 0 or 1 using the
    operations AND, OR, and NOT.

80
Examples of Boolean Polynomials
1.4.80
  • When working with Boolean polynomials, we must
    first know the specific input variables.
  • Examples
  • f(x,y,z) x y z
  • f(x,y) x xy
  • f(x,y,z) x(y z)

81
Evaluating Boolean Polynomials
1.4.81
  • Using x y x y (x y) xy
  • 1 1 0 0 1 1
  • 1 0 0 1 1 0
  • 0 1 1 0 1 0
  • 0 0 1 1 0 1
  • Examples Find f(x,y) x xy
  • x y x xy (x xy)
  • 1 1 0 1 1
  • 1 0 0 0 0
  • 0 1 1 0 1
  • 0 0 1 0 1

82
Normal Forms
1.4.82
  • Expressing a Boolean Polynomial in its normal
    form provides an easy method to calculate its
    truth table.
  • We can create two different normal forms for
    Boolean Polynomials the disjunctive and the
    conjunctive normal form.
  • These forms are made up of special terms called
    minterms or maxterms.

83
Disjunctive Normal Form
1.4.83
  • A minterm is a Boolean polynomial that is only
    the product of each variable or its negation (but
    not both).
  • Examples f(x,y) xy f(x,y,z)
    xyz f(w,x,y,z) wxyz
  • The disjunctive normal form (DNF) is a Boolean
    polynomial that is the sum of minterms (sum of
    products).

84
Disjunctive Normal Form (contd.)
1.4.84
  • Express f(x,y,z) x xz in its DNF.
  • f(x,y,z) x xz
  • x(y y)(z z) x(y y)z
  • (xy xy)(z z) (xy xy)z
  • xyz xyz xyz xyz xyz xyz
  • The thing to note here is that each minterm has
    an output of 1 at only a single, particular line
    of the truth table.
  • i.e. xyz 1 at 101 and 0 elsewhere.

85
Disjunctive Normal Form (contd.)
1.4.85
  • We can now think of the inputs, in fact, as their
    associated minterms to get outputs
  • x y z x y z f(x,y,z)
  • 1 1 1 x y z 1
  • 1 1 0 x y z 1
  • 1 0 1 x yz 1
  • 1 0 0 x yz 1
  • 0 1 1 xy z 1
  • 0 1 0 xy z 0
  • 0 0 1 xyz 1
  • 0 0 0 xyz 0

86
Designing Circuits with Specified Conditions
1.4.86
  • In the other direction
  • x y z f(x,y,z) 1 1 1 0 0 0 0
    0 1 1 0 0 0 0 0 0 1 0 1
    1 1 0 0 0 1 0 0 0 0 0
    0 0 0 1 1 1 0 1 0
    0 0 1 0 1 0 0 1 0 0 0 1
    1 0 0 0 1 0 0 0 0 0 0 0
    0
  • f(x,y,z) xyz xyz xyz xyz

87
Conjunctive Normal Form
1.4.87
  • In a similar fashion, we can analyze functions
    using the conjunctive normal form - the product
    of sums.
  • In this case, we look for the 0s in the
    functions output and associate each with a
    maxterm, whose output is 0 at that row.

88
Equivalent Circuits
1.4.88
  • Two logical circuits are equivalent if and only
    if they have the same truth table.
  • This can be thought similarly as holding when the
    two circuits have the same disjunctive
    (conjunctive) normal form.

89
Section 5.2
5.2.89
  • Properties of sets
  • Methods to show one set is a subset of another
  • Set identities
  • Methods to show two sets are equal

90
Some Subset Relations
5.2.90
  • For all sets A, B, and C
  • 1. A Ç B Í A and A Ç B Í B
  • 2. A Í A È B and B Í A È B
  • 3. If A Í B and B Í C, then A Í C.

91
Procedural Versions of the Set Operations
5.2.91
  • x Î A È B means x Î A or x Î B.
  • x Î A Ç B means x Î A and x Î B.
  • x Î A - B means x Î A and x Ï B.
  • x Î Ac means x Ï A.
  • (x, y) Î A B means x Î A and y Î B.

92
Example Show A Ç B Í A
5.2.92
  • Let x Î A Ç B. Show x Î A.
  • x Î A Ç B means x Î A and x Î B.
  • In particular, this means x Î A.
  • Hence, given x Î A Ç B, we deduce that x Î A.
  • Therefore A Ç B Í A.

93
Set Identities
5.2.93
  • Commutative Laws
  • A Ç B B Ç A and A È B B È A
  • Associative Laws
  • (A Ç B) Ç C A Ç (B Ç C)
  • (A È B) È C A È (B È C)
  • Distributive Laws
  • A È (B Ç C) (A È B) Ç (A È C)
  • A Ç (B È C) (A Ç B) È (A Ç C)

94
Set Identities (contd.)
5.2.94
  • Intersection with U
  • A Ç U A
  • Universal Bound
  • A È U U
  • Double Complement Law
  • (Ac)c A
  • Idempotent Laws
  • A È A A and A Ç A A

95
Set Identities (contd.)
5.2.95
  • DeMorgans Laws
  • (A Ç B)c Ac È Bc and (A È B)c Ac Ç Bc
  • Set Difference Law
  • A - B A Ç Bc
  • Absorption Laws
  • A È (A Ç B) A
  • A Ç (A È B) A

96
Basic Method to Show Set Equality
5.2.96
  • Let sets A and B be given. Show A B.
  • First, show A Í B.
  • Second, show B Í A.
  • If the Í holds in both directions, then we can
    conclude that A B.

97
Example 1 A È (B Ç C) (A È B) Ç (A È C)
5.2.97
  • First, show A È (B Ç C) Í (A È B) Ç (A È C).
  • Then, show (A È B) Ç (A È C) Í A È (B Ç C).

98
Example 2 If A Í B, thenA È B B and A Ç B A
5.2.98
  • First, show A Ç B Í A.
  • Then, show A Í A Ç B.

99
(A È B) - C (A - C) È (B - C)
5.2.99
  • To show these sets are equal, we will simply
    apply the Properties of Sets.
  • (A È B) - C
  • (A È B) Ç Cc
  • (A Ç Cc) È (B Ç Cc )
  • (A - C) È (B - C )

100
Chapter 2. The Logic ofQuantified Statements
2.1.100
  • Predicates
  • Quantified Statements
  • Valid Arguments and Quantified Statements

101
Section 1. Predicates andQuantified Statements I
2.1.101
  • In Chapter 1, we studied the logic of compound
    statements, but the argument reasoning in there
    cannot show the validity of the following simple
    argument
  • All men are mortal.
  • Socrates is a man.
  • Therefore, Socrates is mortal.

102
Predicates
2.1.102
  • To study these types of logical arguments, we
    turn to predicate calculus.
  • A predicate is a sentence that contains a finite
    number of variables and becomes a statement when
    specific values are substituted for the
    variables.
  • The domain of a predicate variable is the set of
    all values that may be substituted in place of
    the variable.

103
Predicate Notation
2.1.103
  • If P(x) is a predicate and x has a domain D, the
    truth set of P(x) is the set of all elements of D
    that make P(x) true when substituted for x.
  • The truth set is denoted x Î D P(x).
  • If P(x) and Q(x) are predicates and the common
    domain of x is D, then the notation P(x) Þ Q(x)
    denotes that the truth set of P(x) is a subset of
    the truth set of Q(x).
  • If P(x) and Q(x) have the same truth set, we
    denote this as P(x) Û Q(x).

104
The Universal Quantifier
2.1.104
  • We often find predicates involved when we are
    making claims about properties that some or all
    the elements of a set obey. This leads us to look
    at statements using one of two quantifiers.
  • The Universal Quantifier If P(x) is a predicate
    over a domain D, we say a universal statement is
    one of the form "x Î D, P(x).
  • This universal statement is true provided P(x) is
    true for every x in D.
  • Any x Î D with P(x) false, is a counterexample.

105
Examples
2.1.105
  • Example 1 Let D 1,2,3,4,5 and let P(x) be
    the predicate x2 ³ x. Using the Method of
    Exhaustion, we find that 12 ³ 1, 22 ³ 2, 32 ³ 3,
    42 ³ 4, and 52 ³ 5 are all true, hence the
    universal statement "x Î 1,2,3,4,5, x2 ³ x is
    true.
  • Example 2 If we change this universal statement
    to "x Î R, x2 ³ x, it is no longer true since x
    1/2 is a counterexample.

106
The Existential Quantifier
2.1.106
  • The Existential Quantifier If P(x) is a
    predicate over a domain D, we say an existential
    statement is one of the form x Î D ' P(x).
  • This existential statement is true provided P(x)
    is true for at least one x in D, and is false if
    P(x) is false for every x in D.
  • From this, we see that the negation of an
    existential statement is a universal statement,
    and, likewise, the negation of a universal
    statement is an existential one.

107
More Examples
2.1.107
  • Consider x Î D ' x2 lt 0.
  • Example 1 If D C (the Complex numbers), then x
    i yields i2 (-1) lt 0, hence the existential
    statement is true.
  • Example 2 If D R, then by the properties of R,
    we know that x2 ³ 0 for all x in R, hence the
    existential statement is false.
  • This second example show us the negation of x
    Î R ' x2 lt 0 is the universal statement "x Î R,
    x2 ³ 0.

108
Negations of Quantifiers
2.1.108
  • As seen in the previous example, the negation of
    an existential statement is a universal
    statement.
  • Formally, we denote x Î D ' P(x) º "x Î D,
    P(x).
  • By the same process, we have that "x Î D,
    P(x) º x Î D ' P(x).
  • Intuitively, the first says the opposite of at
    least one thing satisfying a property is that
    none do, and the opposite of all things
    satisfying the property is that at least one does
    not.

109
Examples of Negations
2.1.109
  • The negation of Some people are sad.is All
    people are not sad.
  • The negation of All integers are
    rational.is At least one integer is
    irrational.
  • Which of each pair is true?

110
Universal Conditional
2.1.110
  • The statement "x, if P(x), then Q(x)is
    called the universal conditional.
  • Many mathematical statements are universal
    conditionals.
  • Example "x Î R, if x gt 2 then x2 gt 4 (formal)is
    equivalent to (informally)
  • Every real number greater than 2 has a square
    greater than 4.
  • The square of any real number greater than 2 is
    greater than 4.

111
Negation of Quantified Conditionals
2.1.111
  • Since we see the properties of symbolic logic
    carry over when dealing with quantified logic, we
    deduce that "x Î D, if P(x), then Q(x)is
    x Î D ' P(x) and Q(x).
  • Similarly, x Î D ' if P(x), then Q(x)is "x
    Î d, P(x) and Q(x).
  • Negate 1. Every CS student studies CMSC203. 2.
    Some CS students study CMSC203.

112
Section 2 - More Quantified Statements
2.2.112
  • Statements with multiple quantifiers
  • Negations of multiply quantified statements
  • Equivalent forms of universal conditionals.

113
Multiply Quantified Statements
2.2.113
  • Consider the following statement Given any real
    number, there is a smaller real number.
  • This is equivalent to the formal statement "
    xÎR, yÎR ' y lt x.
  • This is an example of a multiply quantified
    statement.

114
Examples
2.2.114
  • The formal statement xÎR ' " yÎR, y lt
    xcan be interpreted informally as
  • There is a non-negative real number with the
    property that all other non-negative real numbers
    are smaller than this number
  • There is a non-negative real number that is
    larger than all other non-negative real numbers.

115
Another Example
2.2.115
  • INFORMAL Everybody loves somebody.
  • FORMAL " people x, a person y ' x loves y.
  • INFORMAL Somebody loves everybody.
  • FORMAL a person x ' " people y, x loves y.

116
Negation of Universal Existentials
2.2.116
  • What is the negation of the statement " people
    x, a person y such that x loves y?
  • Recall this is Everybody loves somebody, so its
    negation would be the case of Somebody who does
    not love anybody.
  • In formal terms a person x ' " people y, x
    does not love y.
  • Thus " x, y ' P(x,y) º x ' " y, P(x,y)

117
Negation of Existential Universals
2.2.117
  • What is the negation of the statement a
    person x such that " people y, x loves y?
  • Recall this is Somebody loves everybody, so its
    negation would be the case of Everybody has at
    least one person they do not love.
  • In formal terms " people x, person y ' x does
    not love y.
  • Thus x ' " y, P(x,y) º " x, y ' P(x,y)

118
Equivalent Forms of Universal Conditionals
2.2.118
  • Given the statement " xÎD, if P(x), then
    Q(x)analogous to our definitions from
    propositional calculus, we can construct the
    following.
  • Contrapositive " xÎD, if Q(x), then P(x).
  • Converse " xÎD, if Q(x), then P(x).
  • Inverse " xÎD, if P(x), then Q(x).
  • Negation xÎD ' P(x), and Q(x).

119
Example
2.2.119
  • Statement " xÎR, if x gt 2, then x2 gt 4.
  • Converse " xÎR, if x2 gt 4, then x gt 2.
  • Inverse " xÎR, if x 2, then x2 4.
  • Contrapositive " xÎR, if x2 4, then x 2.
  • Negation xÎR ' x gt 2 and/but x2 4.

120
Section 3 - Valid Arguments
2.3.120
  • Argument Forms
  • Diagrams to Test for Validity
  • Quantified Converse and Inverse Errors
  • Abduction.

121
Universal Instantiation
2.3.121
  • Consider the following statement All men are
    mortal Socrates is a man. Therefore, Socrates
    is mortal.
  • This argument form is valid and is called
    universal instantiation.
  • In summary, it states that if P(x) is true for
    all xÎD and if aÎD, then P(a) must be true.

122
Universal Modus Ponens
2.3.122
  • Formal Version " xÎD, if P(x), then
    Q(x). P(a) for some aÎD. \ Q(a).
  • Informal Version If x makes P(x) true, then x
    makes Q(x) true. a makes P(x) true. \ a makes
    Q(x) true.
  • The first line is called the major premise and
    the second line is the minor premise.

123
Universal Modus Tollens
2.3.123
  • Formal Version " xÎD, if P(x), then
    Q(x). Q(a) for some aÎD. \ P(a).
  • Informal Version If x makes P(x) true, then x
    makes Q(x) true. a makes Q(x) false. \ a makes
    P(x) false.

124
Examples
2.3.124
  • Universal Modus Ponens or Tollens???
  • If a number is even, then its square is even.
  • 10 is even.
  • Therefore, 100 is even.
  • If a number is even, then its square is even.
  • 25 is odd.
  • Therefore, 5 is odd.

125
Using Diagrams to Show Validity
2.3.125
  • Does this diagram portray the argument of the
    second slide?

Mortals
Men
Socrates
126
Modus Ponens in Pictures
2.3.126
  • For all x, P(x) implies Q(x).P(a).Therefore,
    Q(a).

x Q(x)
x P(x)
a
127
A Modus Tollens Example
2.3.127
  • All humans are mortal.Zeus is not
    mortal.Therefore, Zeus is not human.

Zeus
Mortals
Humans
128
Modus Tollens in Pictures
2.3.128
  • For all x, P(x) implies Q(x).Q(a).Therefore,
    P(a).

x Q(x)
a
x P(x)
129
Converse Error in Pictures
2.3.129
  • All humans are mortal.Felix the cat is
    mortal.Therefore, Felix the cat is human.

Mortals
Felix?
Humans
Felix?
130
Inverse Error in Pictures
2.3.130
  • All humans are mortal.Felix the cat is not
    human.Therefore, Felix the cat is not mortal.

Mortals
Felix?
Felix?
Humans
131
Quantified Form of Converseand Inverse Errors
2.3.131
  • Converse Error " x, P(x) implies Q(x). Q(a),
    for a particular a. \ P(a).
  • Inverse Error " x, P(x) implies Q(x). P(a),
    for a particular a. \ Q(a).

132
An Argument with No
2.3.132
  • Major Premise No Naturals are negative.
  • Minor Premise k is a negative number.
  • Conclusion k is not a Natural number.

Negative numbers
Natural numbers
k
133
Abduction
2.3.133
  • Major Premise All thieves go to Pauls Bar.
  • Minor Premise Tom goes to Pauls Bar.
  • Converse Error Therefore, Tom is a thief.
  • Although we cant conclude decisively if Tom is a
    thief or not, if we have further information that
    99 of the 100 people in Pauls Bar are thieves,
    then the odds are that Tom is a thief and the
    converse error is actually valid here.
  • This is called abduction by Artificial
    Intelligence researchers.

134
Chapter 3 - Elementary NumberTheory and Proofs
3.1.134
  • Direct Indirect Proofs
  • Properties of Primes, Integers, Rationals, and
    Reals
  • Divisibility (Unique Factorization Theorem)
  • Modular Forms (Quotient-Remainder Theorem)
  • The Division Euclidean Algorithms.

135
Section 1 - Direct Proof and Counterexample
3.1.135
  • Mathematics is built on the Axiomatic Method.
  • Start with Definitions and Axioms.
  • Use these in valid arguments to demonstrate
    Theorems.
  • Use all of the above to deduce NEW Theorems.
  • Continue ad infinitum.
  • Get paid! (or pass course!)

136
Even and Odd Integers
3.1.136
  • Definition An integer n is even provided there
    exists an integer k such that n 2k.
  • Definition An integer n is odd provided there
    exists an integer k such that n 2k 1.
  • 38 is even since 38 2(19) and 19 is an integer.
  • 417 is odd since 417 2(208) 1 and 208 Î Z.
  • 417 is not even since 417 2(208.5) but 208.5 Ï
    Z.

137
Prime and Composite Integers
3.1.137
  • Definition An integer n is prime if, and only
    if, n gt 1, and for all positive integers r and s,
    if n rs, then r 1 or s 1.
  • Definition An integer n is composite if, and
    only if, n gt 1, and for all positive integers r
    and s, if n rs, then r ¹ 1 and s ¹ 1.
  • Every natural number gt 1 is either prime or
    composite.
  • 2 is the only even prime number.

138
Proving Existential Statements
3.1.138
  • To show There exists an a such that P(a).
  • Demonstrate an Example Prove there is an even
    integer that can be written in two ways as the
    sum of two primes.
  • Proof 10 3 7 5 5.
  • Construct an Example Prove if r,s Î Z, then 4r
    6s is even.
  • Proof Let r,s Î Z. Thus 2r 3s k Î Z, and
    4r 6s 2k, therefore (4r 6s) is even.

139
Proving Universal Statements
3.1.139
  • Most theorems are of the form " xÎD, if P(x),
    then Q(x).
  • If D is a finite set, we can just exhaust over
    each element n to verify that Q(n) holds.
  • Example Prove all n Î 4, 6, 8, 10, 12 can be
    written as the sum of two primes.
  • Proof 4 2 2 6 3 3
  • 8 3 5 10 3 7
  • 12 5 7.

140
Generalizing from theGeneric Particular
3.1.140
  • When it is not feasible to exhaust over each
    element of the domain, we turn to the method of
    generalizing from the generic particular
  • To show that every element of a domain satisfies
    a certain property, suppose x is a particular,
    but arbitrarily chosen element of the domain, and
    show that x satisfies the property.
  • This is the strategy we employ in the method of
    direct proof.

141
Method of Direct Proof
3.1.141
  • Express the statement to be proved in the
    form " xÎD, if P(x), then Q(x) if possible.
    (Often, this is done mentally)
  • Start the proof by supposing that n is a
    particular but arbitrary element of D for which
    P(n) is true. (Suppose nÎD and P(n))
  • Show that the conclusion Q(n) follows from P(n)
    by using definitions, axioms, previously
    established results, and the rules for logical
    inference.

142
Theorem 3.1.1
3.1.142
  • Prove If the sum of two integers is even, then
    so is their difference.
  • Proof Let m and n be any integers with (m n)
    even. This means there is an integer k such that
    (m n) 2k. Now, (m - n) (m n) - 2n 2k -
    2n 2 (k - n) 2p,where k - n p is an
    integer. Thus (m - n) is even. Also, (n - m)
    -(m - n) 2(-p), so (n - m) is also even.
    Therefore, the difference of m and n is even. QED

143
Directions for Writing Proofs
3.1.143
  • Write the statement to be proved.
  • Clearly mark the beginning of your proof with the
    word Proof.
  • Make your proof self-contained
  • Identify each variable used in the body of the
    proof
  • Introduce only necessary variables and notation
  • Use Lemmas to show significant but related ideas.
  • Write proofs in complete (English) sentences.

144
Common Mistakes
3.1.144
  • Arguing from examples
  • Using the same letter to mean different things
  • Jumping to a conclusion
  • Begging the question (i.e. assuming true that
    which you want to prove)
  • Using if when you mean since, hence, thus,
    therefore, hencely, thusly, hereforthwith, etc.

145
Section 2 - Rational Numbers
3.2.145
  • Recall the definition of a Rational Number A
    real number r is rational provided there exist
    integers a and b such that r a/b and b ¹ 0.
  • Theorem Every integer is a rational
    number.Proof Let a be an integer, then a a/1.
    Moreover, 1 is an integer and 1 ¹ 0. Therefore a
    is a rational number. QED

146
Proving Properties of Rationals
3.2.146
  • We will now look at some theorems and corollaries
    (theorems that follow essentially trivially from
    another theorem) about rational numbers.
  • We will rely on the Closure Properties of the
    Integers under , -, and If a,b are integers,
    then (ab), (a-b), (b-a), and ab are also
    integers.
  • We will also use their Zero-Product Property
    If a,b Î Z, with a ¹ 0 and b ¹ 0, then ab ¹ 0.

147
Closure of the Rationals Under
3.2.147
  • Theorem If r, s Î Q, then (r s) Î Q.
  • Proof Let r, s Î Q. Thus a, b, c, d Î Z such
    that r a/b with b ¹ 0 and s c/d with d ¹ 0.
  • Now, (r s) a/b c/d (ad bc)/bd. Since
    a, b, c, d Î Z, we have that (ad bc) Î Z and
    that bd Î Z. Moreover, since b ¹ 0 and d ¹ 0, we
    conclude that bd ¹ 0. Consequently, (r s) is
    the quotient of integers with non-zero
    denominator. Therefore (r s) Î Q. QED

148
A Corollary
3.2.148
  • Corollary Double a rational is rational.
  • Proof Let r s in the previous theorem.

149
Section 3 - Divisibility
3.3.149
  • Definition If n and d are integers and d ¹ 0,
    thenn is divisible by d provided n d k for
    some integer k.
  • Alternatively, we say n is a multiple of d d
    is a factor of n d is a divisor of n d
    divides n (denoted with d n).

150
Properties of Divisibility
3.3.150
  • Divisors of 0 If k is a non-zero integer, thenk
    divides 0 since 0 k 0.
  • Positive Divisors of a Positive NumberIf a and
    b are positive integers and a b, is a b?
  • Yes. Since a b, k Î Z,such that b a k.
    Moreover, 0 lt k, since a and b are, so 1
    k.Thus a a 1 a k b.
  • Therefore a b.
  • Divisors of 1 The only divisors of 1 are 1 and
    -1.

151
Divisibility of Algebraic Terms
3.3.151
  • Let a and b be integers.
  • Does 3 (3a 3b)?
  • Yes, since (3a 3b) 3(a b) and (a b) Î Z.
  • Does 5 10ab?
  • Yes again, since 10ab 5(2ab) and (2ab) Î Z.
  • If m Î Z and m (a b), does m a and m b?
  • No. 2 8 but 2 5 and 2 3.

152
Divisibility and Non-divisibility
3.3.152
  • There is another way to test for divisibilityIf
    d n, there is integer k with n dk, thenk
    (n/d). So, if (n/d) is an integer, then d n.
  • This leads to an easy way to test for
    non-divisibility If (n/d) is not an integer,
    then d cannot divide n.
  • Examples 3 12 since 12/3 4 Î Z. 5 12
    since 12/5 2.4 Ï Z.

153
Proving Properties of Divisibility
3.3.153
  • Theorem Transitivity of DivisibilityFor all
    a,b,c Î Z, if a b and b c, then a c.
  • Proof Let a, b, and c be integers, and assumea
    b and b c. Thus there exist m,n Î Z with b
    ma and c nb.
  • Now, c nb n(ma) (nm)a. Since m,n Î Z,
    we have nm Î Z, therefore a c. QED
  • Example 3 9 and 9 909, therefore 3 909.

154
Divisibility by a Prime
3.3.154
  • Theorem Every positive integer greater than 1 is
    divisible by a prime number.
  • Proof Let n Î Z with n gt 1. Then either n is
    prime or composite. If n is prime, it is
    divisible by itself, and we are done.
  • Now, assume n is composite. Thus there are
    integers (greater than 1) a and b, such that n
    ab. If a is prime, we are done. If not, factor
    a, .... Will we eventually get to a prime factor?

155
Standard Factored Form
3.3.155
  • Definition Given any integer n gt 1, the standard
    factored form of n is an expression of the
    form n (p1)e1 (p2)e2 (p3)e3...(pk)ek,where
    k is a positive integer p1,p2,...,pk are prime
    numbers with p1 lt p2 lt ... lt pk and e1,e2,...,ek
    are positive integers.
  • Example 3300 33 100 3 11 102 22
    3 52 11.

156
Unique Factorization Theorem
3.3.156
  • Theorem Given any integer n gt 1, there exist
    positive integer k prime numbers p1,p2,...,pk
    and positive integers e1,e2,...,ek, with n
    (p1)e1 (p2)e2 (p3)e3...(pk)ek,and any other
    expression of n as a product of prime numbers is
    identical to this except, perhaps, for the order
    in which the factors appears.
  • This is also referred to as the Fundamental
    Theorem of Arithmetic.

157
Fundamental Theorem of Arithmetic
3.3.157
  • Theorem Every positive integer greater than 1
    has a unique factorization as the product of
    primes.
  • Proof (outline)
  • 1. Apply the previous theorem to each composite
    factor encountered.
  • 2. Sort the final listing to get the prime
    factors in increasing (decreasing?) numeric
    order.
  • 3. Rewrite using exponents.

158
Section 4 - The QuotientRemainder Theorem
3.4.158
  • The Quotient-Remainder Theorem
  • Modular Arithmetic (div and mod functions)
  • Proofs Requiring Division into Cases
  • Representations of the Integers.
  • The Parity Theorem

159
Quotient-Remainder Theorem
3.4.159
  • Theorem Given any integer n and a positive
    integer d, there exist unique integers q and r
    such that n dq r, and 0 r lt d.
  • Example If n 27 and d 5, then
    consider 27 0 5 27 27 1 5
    22 27 2 5 17 27 3 5 12 27 4
    5 7 27 5 5 2 here, r 2 and q
    5. 27 6 5 (-3)

160
div and mod Functions
3.4.160
  • Definition Given a nonnegative integer n and a
    positive integer d, n div d the integer
    quotient obtained when n is divided by d n
    mod d the integer remainder obtained when n
    is divided by d.
  • Symbolically, if n and d are positive integersn
    div d q and n mod d r, where n, d, q, and r
    are as described in the Quotient-Remainder
    Theorem.

161
div and mod Examples
3.4.161
  • Consider the previous example of n 27 and d
    5. Since 27 55 2 yields q 5 and r 2,
    we have that 27 div 5 5 27 mod 5 2.
  • More 100 div 10 10 100 mod 10 0
  • 100 div 8 12 100 mod 8 4
  • 10 div 100 0 10 mod 100 10
  • 365 div 7 52 365 mod 7 1

162
Representations of the Integers
3.4.162
  • Recall, we have claimed previously that every
    integer is either even or odd.
  • ConsiderEven ... -10 -8 -6 -4 -2 0 2 4 6 8 10
    ...Odd ... -9 -7 -5 -3 -1 1 3 5 7 9 11 ...
  • We note that all the evens are n 2q 2q 0
    and all the odds are n 2q 1.
  • Moreover, each successive integer alternates
    parity (its mod 2 value).

163
More Representations of Integers
3.4.163
  • If we continue representing integers via the
    Quotient-Remainder Theorem, we observe
  • Modulus Forms
  • 2 2n 2n 1
  • 3 3n 3n 1 3n 2
  • 4 4n 4n 1 4n 2 4n 3
  • ...
  • k kn kn 1 kn 2 ... kn (k-1)

164
Division into Cases
3.4.164
  • Sometimes when proving a theorem, the logical
    flow will fork into different directions, each of
    which need investigation.
  • This is analogous to needing IF THEN ELSE instead
    of just IF THEN in programming flow.
  • An example is the Parity Theorem.
  • Theorem Any two consecutive integers have
    opposite parity.

165
Division into Cases (contd.)
3.4.165
  • Proof Let m be an integer, so its successor is
    (m1). Show m and (m1) have opposite parity.
  • Case 1 (m even) If m is even, there is an
    integer k such that m 2k, hence (m1) 2k 1,
    thus (m1) is odd. So, m even implies (m1) is
    odd.
  • Case 2 (m odd) If m is odd, there is integer k
    such that m 2k 1. Hence (m1) (2k 1)
    1 2k 2 2(k 1), and so (m1) is even. So, m
    odd implies (m1) is even.
  • Therefore, consecutive integers have opposite
    parity. QED

166
The Square of an Odd Integer
3.4.166
  • Theorem If n is an odd integer, (n2 mod 8) 1.
  • Proof Let n be an odd integer, so it has the
    representation modulo 4 of n 4q1 or 4q3.
  • Case 1 Let n 4q1. Thus n2 (4q1)2 16q2
    8q 1 8(2q2 q) 1.
  • Case 2 Let n 4q3. Thus n2 (4q3)2 16q2
    24q 9 16q2 24q 8 1 8(2q2 3q 1)
    1.
  • Therefore, in either case, (n2 mod 8) 1. QED

167
Section 6 - Indirect Argument
3.6.167
  • Method of Proof by Contradiction
  • Method of Proof by Contraposition
  • Examples of Each Method.

168
Proof by Contradiction
3.6.168
  • Instead of the Universal Modus Ponens argument
    form "x, P(x) Q(x) AND P(x) Þ Q(x), a Proof
    by Contradiction (reductio ad absurdum) follows
    the Universal Modus Tollens form "x, P(x)
    Q(x) AND Q(x) Þ P(x).
  • We obtain a contradiction when the conclusion of
    this form is combined with our standard
    assumption in a direct proof the P(x) holds.
  • This differs marginally from the Method of
    Contraposition which proves directly the validity
    of the comtrapositive statement.

169
Method of Proof By Contradiction
3.6.169
  • Suppose the statement to be proved is FALSE
  • Show this supposition leads logically to a
    contradiction (either to the original hypotheses
    or to some other statement of fact)
  • Conclude that the original statement to be proved
    is TRUE.

170
Example No Greatest Integer
3.6.170
  • Theorem There is no greatest integer.
  • Proof (Contradiction) Suppose there is a
    greatest integer N. Thus for every integer k, k
    N.
  • Now, since N is an integer, by closure, (N1)
    is an integer. Thus N 1 N ,hence
    1 0.
  • Therefore, there is no greatest integer. QED

171
Sums of Rationals and Irrationals
3.6.171
  • Theorem The sum of a ra
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