Logic

1
Logic

• Truth Tables, Propositions, Implications

2
Statements

• Logic is the tool for reasoning about the truth
  or falsity of statements.
• Propositional logic is the study of Boolean
  functions
• Predicate logic is the study of quantified
  Boolean functions (first order predicate logic)

3
Arithmetic vs. Logic

• Arithmetic Logic
• 0 false
• 1 true
• Boolean variable statement variable
• form of function statement form
• value of function truth value of statement
• equality of function equivalence of statements

4
Notation

• Word Symbol
• and v
• or w
• implies 6
• equivalent
• not
• not 5
• parentheses ( ) used for grouping terms

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Notation Examples

• English Symbolic
• A and B A v B
• A or B A w B
• A implies B A 6 B
• A is equivalent to B A B
• not A A
• not A 5A

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Statement Forms

• (p v q) and (q v p) are different as statement
  forms. They look different.
• (p v q) and (q v p) are logically equivalent.
  They have the same truth table.
• A statement form that represents the constant 1
  function is called a tautology. It is true for
  all truth values of the statement variables.
• A statement form that represents the constant 0
  function is called a contradiction. It is false
  for all truth values of the statement variables.

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Truth Tables - NOT

• P 5P
• T F
• F T

8
Truth Tables - AND

• P Q PvQ
• T T T
• T F F
• F T F
• F F F

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Truth Tables - OR

• P Q PwQ
• T T T
• T F T
• F T T
• F F F

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Truth Tables - EQUIVALENT

• P Q PQ
• T T T
• T F F
• F T F
• F F T

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Truth Tables - IMPLICATION

• P Q P6Q
• T T T
• T F F
• F T T
• F F T

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Truth Tables - Example

• P true means rain
• P false means no rain
• Q true means clouds
• Q false means no clouds

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Truth Tables - Example

• P Q P6Q P6Q
• rain clouds rain?clouds T
• rain no clouds rain?no clouds F
• no rain clouds no rain?clouds T
• no rain no clouds no rain?no clouds T

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Algebraic rules for statement forms

• Associative rules
• (p v q) v r p v (q v r)
• (p w q) w r p w (q w r)
• Distributive rules
• p v (q w r) (p v q) w (p v r)
• p w (q v r) (p w q) v (p w r)
• Idempotent rules
• p v p p
• p w p p

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Rules (continued)

• Double Negation
• 55p p
• DeMorgans Rules
• 5(p v q) 5p w 5q
• 5(p w q) 5p v 5q
• Commutative Rules
• p v q q v p
• p w q q w p

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Rules (continued)

• Absorption Rules
• p w (p v q) p
• p v (p w q) p
• Bound Rules
• p v 0 0
• p v 1 p
• p w 0 p
• p w 1 1
• Negation Rules
• p v 5p 0
• p w 5p 1

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A Simple Tautology

• P ? Q is the same as 5Q? 5P
• This is the same as asking P?Q 5Q ? 5P
• How can we prove this true?
• By creating a truth table!
• P Q
• T T
• T F
• F T
• F F

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A Simple Tautology (continued)

• Add appropriate columns
• P Q 5P 5Q
• T T F F
• T F F T
• F T T F
• F F T T

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A Simple Tautology (continued)

• Remember your implication table!
• P Q 5P 5Q P?Q
• T T F F T
• T F F T F
• F T T F T
• F F T T T

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A Simple Tautology (continued)

• Remember your implication table!
• P Q 5P 5Q P?Q 5Q?5P
• T T F F T T
• T F F T F F
• F T T F T T
• F F T T T T

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A Simple Tautology (continued)

• Remember your implication table!
• P Q 5P 5Q P?Q 5Q?5P P?Q 5Q?5P
• T T F F T T T
• T F F T F F T
• F T T F T T T
• F F T T T T T
• Since the last column is all true, then the
  original statement
• P?Q 5Q?5P is a tautology
• Note 5Q?5P is the contrapositive of P?Q

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Translation of English

• If P then Q P?Q
• P only if Q 5Q?5P or
• P?Q
• P if and only if Q P Q
• also written as P iff Q

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Translation of English

• P is sufficient for Q P?Q
• P is necessary for Q 5P?5Q or
• Q?P
• P is necessary and sufficient for Q
• P Q
• P unless Q 5Q?P or
• 5P?Q

24
Predicate Logic

• Consider the statement x2 gt 1
• Is it true or false?
• Depends upon the value of x!
• What values can x take on (what is the domain of
  x)?
• Express this as a function S(x) x2 gt 1
• Suppose the domain is the set of reals.
• The codomain (range) of S is a set of statements
  that are either true or false.

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Example

• S(0.9) 0.92 gt 1 is a false statement!
• S(3.2) 3.22 gt 1 is a true statement!
• The function S is an example of a predicate.
• A predicate is any function whose codomain is a
  set of statements that are either true or false.

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Note

• The codomain is a set of statements
• The codomain is not the truth value of the
  statements
• The domain of predicate is arbitrary
• Different predicates can have different domains
• The truth set of a predicate S with domain D is
  the set of the x e D for which S(x) is
  true x e D S(x) is true
• Or simply x S(x)

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Quantifiers

• The phrase for all is called a universal
  quantifier and is symbolically written as œ
• The phrase for some is called an existential
  quantifier and is written as
• Notations for set of numbers
• Reals Integers
• Rationals Primes
• Naturals (nonnegative integers)

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Goldbachs conjecture

• Every even number greater than or equal to 4 can
  be written as the sum of two primes
• Express it as a quantified predicate
• It is unknown whether or not Goldbachs
  conjecture is true. You are only asked to make
  the assertion
• Another example Every sufficiently large odd
  number is the sum of three primes.

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Negating Quantifiers

• Let D be a set and let P(x) be a predicate that
  is defined for x e D, then
• 5(œ(x e D), P(x)) ((x e D), 5P(x))
• and
• 5((x e D), P(x)) (œ(x e D), 5P(x))