Professor: Munehiro Fukuda

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CSS342 Propositions

• Professor Munehiro Fukuda

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Introduction

• Propositional logic is useful in CS to analyze
  such behavior of code portions.
• Proposition a statement that is either true or
  false, but not both
• Logic reasoning whether a consequence of given
  statements is correct, but not whether each
  statement is true.
• Example
• Proposition pAll mathematicians wear sandals. (A
  ? B)
• Proposition qAnyone who wears sandals is an
  algebraist (B ? C)
• From p and qAll mathematicians are algebraists.
  (A ? C)

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Propositions

• Examples
• In math
• The only positive integers that divide 7 are 1
  and 7 itself (true)
• For every positive integer n, there is a prime
  number larger than n (true)
• In history
• Alfred Hitchcock won an Academy Award in 1940 for
  directing Rebecca (false, he has never won one
  for directing)
• Seattle held the Worlds Fair, Expo 62 (true )
• In programming languages
• Boolean expressions in if-else, while, and for
  statements
• for ( index 0 index lt 100 index )
• .

A proposition
Not a proposition
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Compound propositions

• Conjunction of p and q
• notations p q, p q
• True only if both p and q are true
• truth table
• Disjunction of p or q
• Notations p v q, p q
• True if either p or q or both are true
• truth table

p q p q
F F F
F T F
T F F
T T T
p q p v q
F F F
F T T
T F T
T T T
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Binary Expressions in C Part1

• How do you examine the behavior of if-else?
• if ( a gt 1 a lt 100 )
• true if 1 a 100
• if ( a gt 1 a lt 100 )
• always true

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Negation

• The negation of p
• Notations p, not p, p !p
• Truth table
• Example
• P 1 1 3 (false)
• !p !(1 1 3) 1 1 ? 3 (true)

P !p
F T
T F
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Boolean Algebra (Axioms)

• When T true and F false
• If p ? F then p T, If p ? T then p F
• F F F, T T T
• T T T, F F F
• F T T F F
• F T T F T
• !T F, !F T

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Boolean Algebra (Theorems)

• When T true and F false
• p q q p, p q q p
• (p q) r p (q r), (p q ) r
  p (q r)
• (p q) (p r) p q r, p q
  p r p (q r)
• p F F, p T T
• p T p, p F p
• p !p F, p !p T
• p p p, p p p
• p (p q) p, p p q p
• !(!p) p
• p q !p p q

Commutative Law
Associative Law
Distributive Law
Complement Law
Square Law
Absorption Law
Double Negation
Consensus Law
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Binary Expressions in C Part2

• How do you examine the behavior of if-else
  statements?
• Given two distance objects, d1 and 2, if ( d1 lt
  d2 )

p d1.feet lt d2.feet q d1.feet d2.feet a
d1.inches lt d2.inches b d1.inches
d2.inches c d1.inches gt d2.inches
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Binary Expressions in C Part2(Contd)

• p d1.feet lt d2.feet
• q d1.feet d2.feet
• a d1.inches lt d2.inches
• b d1.inches d2.inches
• c d1.inches gt d2.inches
• p a p b p c q a Distributive
  Law
• p (a b c ) q a !a b c
• p (a !a ) q a Complement Law
• p T q a Theorem 6
• p q a
• ( d1.feet lt d2.feet ) ( d1.feet d2.feet )
  (d1.inches lt d2.inches)

bool Distanceoperator lt ( const Distance d2 )
const return ( feet lt d2.feet ) (feet
d2.feet) (inches lt d2.inches)
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Conditional Propositions

• Given two propositions such as p and q,
• If p then q or p ? q
• is called a conditional proposition.
• P hypothesis, antecedent, or sufficient
  condition
• Q conclusion, consequent, or necessary condition
• if p then q p only if q called logically
  equivalent.
• John may take CSS342 only if he advances to
  CSS343.
• If John takes CSS342, he advances to CSS343.

q
p
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Truth Table of Conditional Propositions

• Chair statement if CSS gets an additional
  80,000, it will hire one new faculty member.
• P CSS gets an additional 80,000.
• Q CSS will hire one new faculty member.
• If both p and q are true, the chair said a
  correct statement.
• If p is true but q is false, the chair said a
  wrong statement.
• If p is false, the chair is not responsible for
  his statement. We should regard it as true.

p q p ? q
T T T
T F F
F T T
F F T
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Binary Expressions in C Part3

• Two propositions
• P 1 gt 2 Q 4 lt 8
• According to the truth table,
• p ? q is true, while q ? p is false.
• C program
• Whats the execution result?
• Linux
• 1 gt 2 -gt 4 lt 8 8
• 4 lt 8 -gt 1 lt 2 0
• Goodall
• 1 gt 2 -gt 4 lt 8 0
• 4 lt 8 -gt 1 lt 2 0
• It depends on how result1 was
• initialized.
• In math, we consider all initialized in true.

void main( ) bool result1 if ( 1 gt 2 )
result1 4 lt 8 cout ltlt "1 gt 2 -gt 4 lt 8 " ltlt
result1 ltlt endl bool result2 if ( 4 lt 8 )
result2 1 gt 2 cout ltlt "4 lt 8 -gt 1 gt 2 "
ltlt result2 ltlt endl
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Logical Equivalence

• If two different compound propositions have the
  same truth values no matter what truth values
  their constituent propositions have, they are
  called
• logically equivalent
• Example
• !(p q) !p !q (De Morgans Laws)

p q p q !(p q) !p !q !p !q
T T T F F F F
T F T F F T F
F T T F T F F
F F F T T T T
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De Morgans Laws

• !(p q) !p !q
• Proved by the truth page on the previous page.
• !(p q) !p !q

p q p q !(p q) !p !q !p !q
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T
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Biconditional Propositions

• Given two propositions such as p and q,
• p if and only if q p iff q or p ? q
• is called a conditional proposition.
• P a necessary and sufficient condition for Q
• Q a necessary and sufficient condition for P

p q p ? q
T T T
T F F
F T F
F F T
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Biconditional Propositions Proof of the Truth
Table
p q p ? q q ? p (p ? q) (q ? p) p ? q
T T T T T T
T F F T F F
F T T F F F
F F T T T T
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Contrapositive

• !q ?!p
• The contrapositive of p ? q
• Logically equivalent

p q p ? q !q !p !q ?!p
T T T F F T
T F F T F F
F T T F T T
F F T T T T