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