Propositional Logic

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Propositional Logic

• CS 1050
• (Rosen Section 1.1, 1.2)

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Proposition

• A proposition is a statement that is either true
  or false, but not both.
• Atlanta was the site of the 1996 Summer Olympic
  games.
• 11 2
• 31 5
• What will my CS1050 grade be?

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Definition 1. Negation of p

• Let p be a proposition. The statement It is
  not the case that p is also a proposition,
  called the negation of p or p (read not p)

p The sky is blue. ?p It is not the case that
the sky is blue. ?p The sky is not blue.
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Definition 2. Conjunction of p and q

• Let p and q be propositions. The proposition p
  and q, denoted by p?q is true when both p and q
  are true and is false otherwise. This is called
  the conjunction of p and q.

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Definition 3. Disjunction of p and q

• Let p and q be propositions. The proposition p
  or q, denoted by p?q, is the proposition that is
  false when p and q are both false and true
  otherwise.

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Definition 4. Exclusive or of p and q

• Let p and q be propositions. The exclusive or
  of p and q, denoted by p?q, is the proposition
  that is true when exactly one of p and q is true
  and is false otherwise.

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Definition 5. Implication p?q

• Let p and q be propositions. The implication p?q
  is the proposition that is false when p is true
  and q is false, and true otherwise. In this
  implication p is called the hypothesis (or
  antecedent or premise) and q is called the
  conclusion (or consequence).

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Implications

• If p, then q
• p implies q
• if p,q
• p only if q
• p is sufficient for q
• q if p
• q whenever p
• q is necessary for p

• Not the same as the if-then construct used in
  programming languages such as If p then S

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Implications

• How can both p and q be false, and p?q be true?
• Think of p as a contract and q as its
  obligation that is only carried out if the
  contract is valid.
• Example If you make more than 25,000, then you
  must file a tax return. This says nothing about
  someone who makes less than 25,000. So the
  implication is true no matter what someone making
  less than 25,000 does.
• Another example
• p Bill Gates is poor.
• q Pigs can fly.
• p?q is always true because Bill Gates is not
  poor. Another way of saying the implication is
• Pigs can fly whenever Bill Gates is poor which
  is true since neither p nor q is true.

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Related Implications

• Converse of p ? q is q ? p

• Contrapositive of p ? q is
  the proposition ?q ? ?p

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Definition 6. Biconditional

• Let p and q be propositions. The biconditional
  p?q is the proposition that is true when p and q
  have the same truth values and is false
  otherwise. p if and only if q, p is necessary
  and sufficient for q

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Practice
p You learn the simple things well. q The
difficult things become easy.

• The difficult things become easy but you did not
  learn the simple things well.
• You learn the simple things well but the
  difficult things did not become easy.

• You do not learn the simple things well.
• If you learn the simple things well then the
  difficult things become easy.
• If you do not learn the simple things well, then
  the difficult things will not become easy.

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Practice
p You learn the simple things well. q The
difficult things become easy.

• The difficult things become easy but you did not
  learn the simple things well.
• You learn the simple things well but the
  difficult things did not become easy.

• You do not learn the simple things well.
• If you learn the simple things well then the
  difficult things become easy.
• If you do not learn the simple things well, then
  the difficult things will not become easy.

?p
q ? ?p
p?q
p ? ?q
?p ? ?q
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Truth Table Puzzle

• Steve would like to determine the relative
  salaries of three coworkers using two facts (all
  salaries are distinct)
• If Fred is not the highest paid of the three,
  then Janice is.
• If Janice is not the lowest paid, then Maggie is
  paid the most.
• Who is paid the most and who is paid the least?

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p Janice is paid the most. q Maggie is paid
the most. r Fred is paid the most. s Janice is
paid the least. p q r s ?r?p ?s ?q (?r?p)?
(?s?q) T F F F T F F F T F T F
T F F F T T T T T F T F F F T F F F T F T
F F Fred, Maggie, Janice

• If Fred is not the highest paid of the three,
  then Janice is.
• If Janice is not the lowest paid, then Maggie is
  paid the most.

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p Janice is paid the most. q Maggie is paid
the most. r Fred is paid the most. s Janice is
paid the least. p q r s ?r?p s ?q (?r?p)?
(s?q) T F F F T T T F T F T F
T F F F T T T F F F T F F F T F F F T F T
T T Fred, Janice, Maggie or Janice, Maggie,
Fred or Janice, Fred, Maggie

• If Fred is not the highest paid of the three,
  then Janice is.
• If Janice is the lowest paid, then Maggie is paid
  the most.

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Bit Operations
A computer bit has two possible values 0 (false)
and 1 (true). A variable is called a Boolean
variable is its value is either true or
false. Bit operations correspond to the logical
connectives ? OR ? AND ? XOR Information can
be represented by bit strings, which are
sequences of zeros and ones, and manipulated by
operations on the bit strings.
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Truth tables for the bit operations OR, AND, and
XOR
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Logical Equivalence

• An important technique in proofs is to replace a
  statement with another statement that is
  logically equivalent.
• Tautology compound proposition that is always
  true regardless of the truth values of the
  propositions in it.
• Contradiction Compound proposition that is
  always false regardless of the truth values of
  the propositions in it.

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Logically Equivalent

• Compound propositions P and Q are logically
  equivalent if P?Q is a tautology. In other
  words, P and Q have the same truth values for all
  combinations of truth values of simple
  propositions.
• This is denoted P?Q (or by P Q)

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Example DeMorgans

• Prove that ?(p?q) ? (?p ? ?q)
• p q (p?q) ?(p?q) ?p ?q (?p ? ?q)

T T T F F T F F
T F F F F
T F F T F
T F T F F

F T T T T

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Illustration of De Morgans Law
?(p?q)
p
q
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Illustration of De Morgans Law
?p
p
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Illustration of De Morgans Law
?q
q
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Illustration of De Morgans Law
?p ? ?q
p
q
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Example Distribution
Prove that p ? (q ? r) ? (p ? q) ? (p ? r)
p q r q?r p?(q?r) p?q p?r
(p?q)?(p?r) T T T T T T
T T T T F F T
T T T T F T F
T T T T T F F
F T T T T F
T T T T T T
T F T F F F T
F F F F T F F
F T F F F F F F
F F F
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Prove p?q?(p?q) ? (q?p)

• p q p?q p?q q?p (p?q)?(q?p)
• T T T T T T
• T F F F T F
• F T F T F F
• F F T T T T
• We call this biconditional equivalence.

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List of Logical Equivalences
p?T ? p p?F ? p Identity Laws p?T ? T
p?F ? F Domination Laws p?p ? p p?p ? p
Idempotent Laws ?(?p) ? p Double Negation
Law p?q ? q?p p?q ? q?p Commutative
Laws (p?q)? r ? p? (q?r) (p?q) ? r ? p ? (q?r)

Associative Laws
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List of Equivalences
p?(q?r) ? (p?q)?(p?r) Distribution Laws p?(q?r)
? (p?q)?(p?r) ?(p?q)?(?p ? ?q) De Morgans
Laws ?(p?q)?(?p ? ?q) Miscellaneous p ? ?p ?
T Or Tautology p ? ?p ? F And
Contradiction (p?q) ? (?p ? q) Implication
Equivalence p?q?(p?q) ? (q?p) Biconditional
Equivalence