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Discrete Math by R.S. Chang, Dept. CSIE, NDHU 1
Fundamentals of Logic
1. What is a valid argument or proof?
2. Study system of logic
3. In proving theorems or solving problems, creativity and insight are needed, which cannot be taught
Chapter 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 2
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
statements (propositions): declarative sentences that areeither true or false--but not both.
Eg. Margaret Mitchell wrote Gone with the Wind. 2+3=5.
not statements:
What a beautiful morning!Get up and do your exercises.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 3
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tablesprimitive and compound statements
combined from primitive statements by logical connectivesor by negation ( )
logical connectives:
p p,
(a) conjunction (AND): (b) disjunction(inclusive OR):(c) exclusive or: (d) implication: (if p then q)(e) biconditional: (p if and only if q, or p iff q)
p qp q
p qp q
p q
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 4
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
"The number x is an integer." is not a statement because itstruth value cannot be determined until a numerical value is assigned for x.
First order logic vs. predicate logic
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 5
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
Truth Tables
p q p q p q p q p qp q
0 0 0 0 0 1 1
0 1 0 1 1 1 0
1 0 0 1 1 0 0
1 1 1 1 0 1 1
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 6
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
Ex. 2.1 s: Phyllis goes out for a walk.t: The moon is out.u: It is snowing.
( )t u s : If the moon is out and it is not snowing, thenPhyllis goes out for a walk.
If it is snowing and the moon is not out, then Phylliswill not go out for a walk. ( )u t s
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 7
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
Def. 2.1. A compound statement is called a tautology(T0) if it is true for all truth value assignments for its component statements. If a compound statement is false for all such assignments, then it is called a contradiction(F0).
p p q
p p q
( )
( )
: tautology
: contradiction
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 8
Chapter 2. Fundamentals of Logic
2.1 Basic connectives and truth tables
an argument: ( )p p p qn1 2
premises conclusion
If any one of p p pn1 2, , , is false, then no matter whattruth value q has, the implication is true. Consequently,if we start with the premises p p pn1 2, , , --each with
truth value 1--and find that under these circumstancesq also has value 1, then the implication is a tautology andwe have a valid argument.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 9
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Ex. 2.7
0011
0101
1100
1101
1101
p q p p q p q
Def 2.2 . logically equivalent
s s1 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 10
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
logically equivalent
)()(
)()()(
)()()(
)()()(
)(
qpqp
qpqpqp
pqqpqp
pqqpqp
qpqp
We can eliminate the connectives and
from compound statements.(and,or,not) is a complete set.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 11
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Ex 2.8. DeMorgan's Laws
( )
( )
p q p q
p q p q
p and q can be any compound statements.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 12
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic( )
( ) ( )
( )
( )
( ) ( ) ( )
( ) ( )
1
2
3
4
p p
p q p q
p q p q
p q q p
p q q p
p q r p q r
p q r p q r
Law of Double Negation
Demorgan's Laws
Commutative Laws
Associative Laws
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 13
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ,
( ) ,
( ) ,
( ) ,
( ) ( ) , ( )
5
6
7
8
9
10
0 0
0 0
0 0 0 0
p q r p q p r
p q r p q p r
p p p p p p
p F p p T p
p p T p p F
p T T P F F
p p q p p p q p
Distributive Law
Idempotent Law
Identity Law
Inverse Law
Domination Law
Absorption Law
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 14
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
All the laws, aside from the Law of Double Negation, allfall naturally into pairs.
Def. 2.3 Let s be a statement. If s contains no logical connectivesother than and , then the dual of s, denoted sd, is thestatement obtained from s by replacing each occurrence ofand by and , respectively, and each occurrence of T0 and F0 by F0 and T0, respectively.
Eg. s p q r T s p q r Fd: ( ) ( ), : ( ) ( ) 0 0
The dual of is p q ( ) p q p qd
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 15
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Theorem 2.1 (The Principle of Duality) Let s and t be statements.If , then .s t s td d
Ex. 2.10 P p q p q: ( ) ( ) is a tautology. Replace
each occurrence of p by r sP r s q r s q1: [( ) ] [ ( ) ] is also a tautology.
First Substitution Rule (replace each p by another statement q)
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 16
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Second Substitution Rule
Ex. 2.11 P p q r
P p q r
: ( )
: ( )
1
Then, P P 1
because ( ) ( )p q p q
Ex. 2.12 Negate and simplify the compound statement ( )p q r
[( ) ] [ ( ) ]
[( ) ] ( )
( )
p q r p q r
p q r p q r
p q r
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 17
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Ex. 2.13 What is the negation of "If Joan goes to Lake George,then Mary will pay for Joan's shopping spree."?
Because ( ) ( )p q p q p q
The negation is "Joan goes to Lake George, but (or and)Mary does not pay for Joan's shopping spree."
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 18
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Ex. 2.15
p q0011
1101
1101
1011
1011
0101
p q q p q p p q
contrapositive of p q
converseinverse
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 19
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
Compare the efficiency of two program segments.z:=4;for i:=1 to 10 do begin x:=z-1; y:=z+3*i; if ((x>0) and (y>0)) then writeln(‘The value of the sum x+y is’, x+y) end
.
.
.if x>0 then if y>0 then …
rqp )( )( rqp
Number of comparisons?20 vs. 10+3=13
logically equivalent
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 20
Chapter 2. Fundamentals of Logic
2.2 Logical Equivalence: The Laws of Logic
simplification of compound statement
Ex. 2.16
pFp
qqp
qpqp
qpqp
qpqp
0
)(
)()(
)()(
)()( Demorgan's Law
Law of Double Negation
Distributive Law
Inverse Law andIdentity Law
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 21
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
an argument: ( )p p p qn1 2
premises conclusion
is a valid argument
( )p p p qn1 2 is a tautology
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 22
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of InferenceEx. 2.19 statements: p: Roger studies. q: Roger plays tennis.r: Roger passes discrete mathematics.
premises: p1: If Roger studies, then he will pass discrete math.p2: If Roger doesn't play tennis, then he'll study.p3: Roger failed discrete mathematics.Determine whether the argument ( )p p p q1 2 3 is
valid.p p r p q p p r
p p p q
p r q p r q
1 2 3
1 2 3
: , : , :
( )
[( ) ( ) ]
which is a tautology,the original argumentis true
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 23
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
p r s0 0 0 0 1 1 10 0 1 0 1 1 10 1 0 0 1 0 10 1 1 0 1 1 11 0 0 0 1 1 11 0 1 0 1 1 11 1 0 1 0 0 11 1 1 1 1 1 1
p r ( )p r s r s [ (( ) )] ( )p p r s r s Ex. 2.20
a tautologydeduced or inferred fromthe two premises
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 24
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Def. 2.4. If p, q are any arbitrary statements such thatis a tautology, then we say that p logically implies q and we
p q
p q to denote this situation.write
p q means p q is a tautology.
p q means p q is a tautology.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 25
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
rule of inference: use to validate or invalidate a logicalimplication without resorting to truth table (which will beprohibitively large if the number of variables are large)
Ex 2.22 Modus Ponens (the method of affirming) or the Rule of Detachment
[ ( )]p p q q p
p q
q
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 26
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Example 2.23 Law of the Syllogism
[( ) ( )] ( )p q q r p r
p q
q r
p r
Ex 2.25 p
p q
q r
r
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 27
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex. 2.25 Modus Tollens (method of denying)
[( ) ]p q q p p q
q
p
p r
r s
t s
t u
u
p
p s
p
example:
s tt u
s u
p u
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 28
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex. 2.25 Modus Tollens (method of denying)
[( ) ]p q q p p q
q
p
p r
r s
t s
t u
u
p
ts
p s
p
example:
another reasoning
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 29
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
fallacy p q
q
p
(1) If Margaret Thatcher is the president of the U.S., then she is at least 35 years old.(2) Margaret Thatcher is at least 35 years old.(3) Therefore, Margaret Thatcher is the president of the US.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 30
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
p q
p
q
fallacy
(1) If 2+3=6, then 2+4=6.(2) 2+3(3) Therefore, 2+4
6 6
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 31
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex 2.26 Rule of Conjunctionp
q
p q
Ex. 2.27 Rule of Disjunctive Syllogism
p q
p
q
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 32
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex. 2.28 Rule of Contradiction
p F
p0
Proof by Contradiction
( )p p p qn1 2 To prove
we prove 021
021
)(
))((
Fqppp
Fqppp
n
n
qp
qp
qp
)(
)(
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 33
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex. 2.29 p r
p q
q s
r s
r p r q
r s
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 34
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex. 2.30p q
q r s
r t u
p t
u
( )
( )
p, t
qr, s
t u
u
No systematic way to prove except by truth table (2n).
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 35
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex 2.32 Proof by Contradiction
p q
q r
r
p
p q
q r
r
p
F0
qr
F0
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 36
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
p
p
p
q rn
1
2
p
p
p
q
r
n
1
2
p q r
p q r
p q r
p q r
p q r
p q r
( )
( )
( )
( )
( )
( )
reasoning
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 37
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
Ex 2.33u r
r s p t
q u s
t
q p
( ) ( )
( )
u r
r s p t
q u s
t
q
p
( ) ( )
( ) u s u, s
r p t
p
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 38
Chapter 2. Fundamentals of Logic
2.3 Logical Implication: Rules of Inference
How to prove that an argument is invalid?Just find a counterexample (of assignments) for it !
Ex 2.34 Show the following to be invalid.p
p q
q r s
t r
s t
( )
0
1
s=0,t=1
p=1
r=1 ( )r s 0
q=0
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 39
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Def. 2.5 A declarative sentence is an open statement if(1) it contains one or more variables, and(2) it is not a statement, but(3) it becomes a statement when the variables in it are replaced by certain allowable choices.
examples: The number x+2 is an even integer. x=y, x>y, x<y, ...
universe
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 40
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
notations: p(x): The number x+2 is an even integer.
q(x,y): The numbers y+2, x-y, and x+2y are even integers.
p(5): FALSE, p( )7 : TRUE, q(4,2): TRUE
p(6): TRUE, p( )8 : FALSE, q(3,4): FALSE
For some x, p(x) is true.For some x,y, q(x,y) is true.
For some x, is true.For some x,y, is true.
p x( )q x y( , )
Therefore,
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 41
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
existential quantifier: For some x:universal quantifier: For all x:
x
x
x in p(x): free variablex in : bound variablex p x, ( ) x p x, ( ) is either
true or false.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 42
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Ex 2.36
p x x
q x x
r x x x
s x x
( ):
( ):
( ):
( ):
0
0
3 4 0
3 0
2
2
2
x p x r x TRUE
x p x q x TRUE
x p x q x TRUE
x q x s x FALSE
x r x s x FALSE
x r x p x FALSE
[ ( ) ( )]:
[ ( ) ( )]:
[ ( ) ( )]:
[ ( ) ( )]:
[ ( ) ( )]:
[ ( ) ( )]:
x=4
x=1
x=5,6,...
x=-1
universe: real numbers
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 43
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Ex 2.37 implicit quantification
sin cos2 2 1x x x x x(sin cos )2 2 1is
"The integer 41 is equal to the sum of two perfect squares."is m n m n[ ]41 2 2
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 44
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Def. 2.6 logically equivalent for open statement p(x) and q(x)
x p x q x[ ( ) ( )]
p(x) logically implies q(x)
x p x q x[ ( ) ( )]
, i.e., p x q x( ) ( ) for any x
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 45
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Ex. 2.42 Universe: all integers
r x x
s x x
( ):
( ):
2 1 5
92
then x r x s x[ ( ) ( )] is false
but xr x xs x( ) ( ) is true
Therefore, x r x s x[ ( ) ( )] xr x xs x( ) ( )
but x p x q x xp x xq x[ ( ) ( )] [ ( ) ( )]
for any p(x), q(x) and universe
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 46
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
For a prescribed universe and any open statements p(x), q(x):
x p x q x xp x xq x
x p x q x xp x xq x
x p x q x xp x xq x
xp x xq x x p x q x
[ ( ) ( )] [ ( ) ( )]
[ ( ) ( )] [ ( ) ( )]
[ ( ) ( )] [ ( ) ( )]
[ ( ) ( )] [ ( ) ( )] Note this!
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 47
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
How do we negate quantified statements that involve a singlevariable?
[ ( )] ( )
[ ( )] ( )
[ ( )] ( )
[ ( )] ( )
xp x x p x
xp x x p x
x p x xp x
x p x xp x
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 48
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Ex. 2.44 p(x): x is odd.q(x): x2-1 is even.
Negate x p x q x[ ( ) ( )] (If x is odd, then x2-1 is even.)
[ ( ( ) ( )] [ ( ( ) ( ))]
[ ( ( ) ( ))] [ ( ) ( )]
x p x q x x p x q x
x p x q x x p x q x
There exists an integer x such that x is odd and x2-1 is odd.(a false statement, the original is true)
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 49
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
multiple variables
x yp x y y xp x y
x yp x y y xp x y
( , ) ( , )
( , ) ( , )
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 50
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
BUT
Ex. 2.48 p(x,y): x+y=17.
x yp x y( , ) : For every integer x, there exists an integer y such that x+y=17. (TRUE)
y xp x y( , ) : There exists an integer y so that for all integer x, x+y=17. (FALSE)
),(),( yxxpyyxypx Therefore,
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 51
Chapter 2. Fundamentals of Logic
2.4 The Use of Quantifiers
Ex 2.49
[ [( ( , ) ( , )) ( , )]]
[ [( ( , ) ( , )) ( , )]]
[( ( , ) ( , )) ( , )]
[ [ ( , ) ( , )] ( , )]
[( ( , ) ( , )) ( , )]
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
Discrete Math by R.S. Chang, Dept. CSIE, NDHU 52
Chapter 2. Fundamentals of Logic
Exercises. Exercise 2.2: 20 Exercise 2.3: 10 Exercise 2.4: 18, 26