Logic
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Logic

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Logic Logic Presentation Transcript

  • LOGIC
  • 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)
  • 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
  • Notation Word Symbol and v or w implies 6 equivalent ] not ~ not 5 parentheses ( ) used for grouping terms
  • 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
  • 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.
  • Truth Tables - NOT P 5P T F F T
  • Truth Tables - AND P Q PvQ T T T T F F F T F F F F
  • Truth Tables - OR P Q PwQ T T T T F T F T T F F F
  • Truth Tables - EQUIVALENT P Q P]Q T T T T F F F T F F F T
  • Truth Tables - IMPLICATION P Q P6Q T T T T F F F T T F F T
  • Truth Tables - Example P true means rain P false means no rain Q true means clouds Q false means no clouds
  • 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
  • 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
  • Rules (continued) • Double Negation: 55p ] p • DeMorgan’s 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Predicate Logic • Consider the statement: x2 > 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 > 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.
  • Example • S(0.9) = 0.92 > 1 is a false statement! • S(3.2) = 3.22 > 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.
  • 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 ε D for which S(x) is true: {x ε D | S(x) is true} • Or simply: {x | S(x)}
  • 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)
  • Goldbach’s 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 Goldbach’s conjecture is true. You are only asked to make the assertion • Another example: Every sufficiently large odd number is the sum of three primes.
  • Negating Quantifiers • Let D be a set and let P(x) be a predicate that is defined for x ε D, then 5(œ(x ε D), P(x)) ] (›(x ε D), 5P(x)) and 5(›(x ε D), P(x)) ] (œ(x ε D), 5P(x))