Title: Chapter 5 - Set Theory
1Chapter 5 - Set Theory
5.1.1
- 1. Basic Definitions
- 2. Empty Set, Partitions, Power Set
- 3. Properties of Sets
2Section 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
35.1.3
- We use ellipses to simplify things
- 1, 2, 3, , 10
- 1, 2, 3,
- , -2, -1, 0, 1, 2,
- Be careful! (1, 2, ???)
45.1.4
- 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.
5Special 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.
6Natural 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.
7The 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.
8The 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.
9The 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
10The 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 .
11The 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.
12The 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.
13Subsets
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.
14Examples 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.
15Set 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.
16Operations 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.
17Examples 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, )
18Cartesian 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)
19Generalized 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
20Formal 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.
21Formal 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.
22Formal 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.
23Examples
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,
....
24Section 5.3
5.3.24
- The Empty Set
- Partitions
- Power Sets
- Boolean Algebras
25The 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 .
26Set 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.
27Partitions 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.
28Partitions 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?
29Power 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.
30Boolean 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
31Chapter 1. Symbolic Logic
1.1.31
- Logical Form and Equivalence
- Conditional Statements
- Valid and Invalid Arguments
- Digital Logic Circuits (Boolean Polynomials)
32Logic 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.
33Counterexamples
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)
34Compound 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)
35Compound 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.
36Compound 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.
37Logical 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
38Example (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
39Logical 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.
40Tautology 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?
41Algebra 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)
42Algebra 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
43Algebra 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
44Section 1.2
1.2.44
- Conditional Statements
- Logical Equivalences Involving Conditionals
- Converses, Inverses, and Contrapositives
- Biconditional Statements
45Conditional 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.
46Hypotheses 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.
47Logical 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.
48Negation 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.
49Converse 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.
50Contrapositive 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.
51Inverse 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.
52Equivalent 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.
53Biconditional 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
54Using 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.
55Section 1.3
1.3.55
- Valid and invalid argument forms.
- Special valid argument forms.
- Dilemmas
- Fallacies.
- Contradictions and valid arguments.
56Valid 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.
57Valid 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.
58Testing 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.
59A 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
60An 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
61Special 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.
62Modus 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.
63Disjunctive 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.
64Conjunctive 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.
65Disjunctive 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.
66Hypothetical 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.
67Dilemma 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.
68Application 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.
69Find 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.
70Find 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!
71Fallacies
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.
72Converse 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).
73Inverse 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).
74Contradiction 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.
75Section 1.4
1.4.75
- Digital Logic Circuits
- Boolean Polynomials
- Normal Forms (Disjunctive/Conjunctive)
- Designing Circuits with Specified Conditions
- Showing Two Circuits Are Equivalent
76Digital 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
77Logical 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
78Notation
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
79Boolean 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.
80Examples 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)
81Evaluating 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
82Normal 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.
83Disjunctive 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).
84Disjunctive 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.
85Disjunctive 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
86Designing 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
87Conjunctive 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.
88Equivalent 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.
89Section 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
90Some 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.
91Procedural 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.
92Example 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.
93Set 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)
94Set 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
95Set 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
96Basic 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.
97Example 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).
98Example 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 )
100Chapter 2. The Logic ofQuantified Statements
2.1.100
- Predicates
- Quantified Statements
- Valid Arguments and Quantified Statements
101Section 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.
102Predicates
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.
103Predicate 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).
104The 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.
105Examples
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.
106The 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.
107More 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.
108Negations 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.
109Examples 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?
110Universal 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.
111Negation 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.
112Section 2 - More Quantified Statements
2.2.112
- Statements with multiple quantifiers
- Negations of multiply quantified statements
- Equivalent forms of universal conditionals.
113Multiply 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.
114Examples
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.
115Another 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.
116Negation 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)
117Negation 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)
118Equivalent 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).
119Example
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.
120Section 3 - Valid Arguments
2.3.120
- Argument Forms
- Diagrams to Test for Validity
- Quantified Converse and Inverse Errors
- Abduction.
121Universal 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.
122Universal 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.
123Universal 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.
124Examples
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.
125Using Diagrams to Show Validity
2.3.125
- Does this diagram portray the argument of the
second slide?
Mortals
Men
Socrates
126Modus 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
127A Modus Tollens Example
2.3.127
- All humans are mortal.Zeus is not
mortal.Therefore, Zeus is not human.
Zeus
Mortals
Humans
128Modus 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)
129Converse 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?
130Inverse 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
131Quantified 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).
132An 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
133Abduction
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.
134Chapter 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.
135Section 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!)
136Even 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.
137Prime 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.
138Proving 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.
139Proving 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.
140Generalizing 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.
141Method 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.
142Theorem 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
143Directions 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.
144Common 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.
145Section 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
146Proving 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.
147Closure 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
148A Corollary
3.2.148
- Corollary Double a rational is rational.
- Proof Let r s in the previous theorem.
149Section 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).
150Properties 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.
151Divisibility 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.
152Divisibility 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.
153Proving 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.
154Divisibility 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?
155Standard 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.
156Unique 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.
157Fundamental 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.
158Section 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
159Quotient-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)
160div 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.
161div 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
162Representations 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).
163More 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)
164Division 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.
165Division 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
166The 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
167Section 6 - Indirect Argument
3.6.167
- Method of Proof by Contradiction
- Method of Proof by Contraposition
- Examples of Each Method.
168Proof 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.
169Method 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.
170Example 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
171Sums of Rationals and Irrationals
3.6.171