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Logic: Learning Objectives

Logic: Learning Objectives. Learn about statements (propositions) Learn how to use logical connectives to combine statements Explore how to draw conclusions using various argument forms Become familiar with quantifiers and predicates CS Boolean data type If statement Impact of negations

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Logic: Learning Objectives

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  1. Logic: Learning Objectives • Learn about statements (propositions) • Learn how to use logical connectives to combine statements • Explore how to draw conclusions using various argument forms • Become familiar with quantifiers and predicates • CS • Boolean data type • If statement • Impact of negations • Implementation of quantifiers Discrete Mathematical Structures: Theory and Applications

  2. Mathematical Logic • Definition: Methods of reasoning, provides rules and techniques to determine whether an argument is valid • Theorem: a statement that can be shown to be true (under certain conditions) • Example: If x is an even integer, then x + 1 is an odd integer • This statement is true under the condition that x is an integer is true Discrete Mathematical Structures: Theory and Applications

  3. Mathematical Logic • A statement, or a proposition, is a declarative sentence that is either true or false, but not both • Lowercase letters denote propositions • Examples: • p: 2 is an even number (true) • q: 3 is an odd number (true) • r: A is a consonant (false) • The following are not propositions: • p: My cat is beautiful • q: Are you in charge? Discrete Mathematical Structures: Theory and Applications

  4. Mathematical Logic • Truth value • One of the values “truth” or “falsity” assigned to a statement • True is abbreviated to T or 1 • False is abbreviated to F or 0 • Negation • The negation of p, written ∼p, is the statement obtained by negating statement p • Truth values of p and ∼p are opposite • Symbol ~ is called “not” ~p is read as as “not p” • Example: • p: A is a consonant • ~p: it is the case that A is not a consonant • q: Are you in charge? Discrete Mathematical Structures: Theory and Applications

  5. Mathematical Logic • Truth Table • Conjunction • Let p and q be statements.The conjunction of p and q, written p ^ q , is the statement formed by joining statements p and q using the word “and” • The statement p∧q is true if both p and q are true; otherwise p∧q is false Discrete Mathematical Structures: Theory and Applications

  6. Mathematical Logic • Conjunction • Truth Table for Conjunction: Discrete Mathematical Structures: Theory and Applications

  7. Mathematical Logic • Disjunction • Let p and q be statements. The disjunction of p and q, written p v q , is the statement formed by joining statements p and q using the word “or” • The statement p v q is true if at least one of the statements p and q is true; otherwise p v q is false • The symbol v is read “or” Discrete Mathematical Structures: Theory and Applications

  8. Mathematical Logic • Disjunction • Truth Table for Disjunction: Discrete Mathematical Structures: Theory and Applications

  9. Mathematical Logic • Implication • Let p and q be statements.The statement “if p then q” is called an implication or condition. • The implication “if p then q” is written p  q • p  q is read: • “If p, then q” • “p is sufficient for q” • q if p • q whenever p Discrete Mathematical Structures: Theory and Applications

  10. Mathematical Logic • Implication • Truth Table for Implication: • p is called the hypothesis, q is called the conclusion Discrete Mathematical Structures: Theory and Applications

  11. Mathematical Logic • Implication • Let p: Today is Sunday and q: I will wash the car. The conjunction p  q is the statement: • p  q : If today is Sunday, then I will wash the car • The converse of this implication is written q  p • If I wash the car, then today is Sunday • The inverse of this implication is ~p  ~q • If today is not Sunday, then I will not wash the car • The contrapositive of this implication is ~q  ~p • If I do not wash the car, then today is not Sunday Discrete Mathematical Structures: Theory and Applications

  12. Mathematical Logic • Biimplication • Let p and q be statements. The statement “p if and only if q” is called the biimplication or biconditional of p and q • The biconditional “p if and only if q” is written p  q • p  q is read: • “p if and only if q” • “p is necessary and sufficient for q” • “q if and only if p” • “q when and only when p” Discrete Mathematical Structures: Theory and Applications

  13. Mathematical Logic • Biconditional • Truth Table for the Biconditional: Discrete Mathematical Structures: Theory and Applications

  14. Mathematical Logic • Statement Formulas • Definitions • Symbols p ,q ,r ,...,called statement variables • Symbols ~, ^, v, →,and ↔ are called logical connectives • A statement variable is a statement formula • If A and B are statement formulas, then the expressions (~A ), (A ^B) , (A v B ), (A → B ) and (A ↔ B ) are statement formulas • Expressions are statement formulas that are constructed only by using 1) and 2) above Discrete Mathematical Structures: Theory and Applications

  15. Mathematical Logic • Precedence of logical connectives is: • ~ highest • ^ second highest • v third highest • → fourth highest • ↔ fifth highest Discrete Mathematical Structures: Theory and Applications

  16. Mathematical Logic • Example: • Let A be the statement formula (~(p v q )) → (q ^p ) • Truth Table for A is: Discrete Mathematical Structures: Theory and Applications

  17. Mathematical Logic • Tautology • A statement formula A is said to be a tautology if the truth value of A is T for any assignment of the truth values T and F to the statement variables occurring in A • Contradiction • A statement formula A is said to be a contradiction if the truth value of A is F for any assignment of the truth values T and F to the statement variables occurring in A Discrete Mathematical Structures: Theory and Applications

  18. Mathematical Logic • Logically Implies • A statement formula A is said to logically imply a statement formula B if the statement formula A → B is a tautology. If A logically implies B, then symbolically we write A → B • Logically Equivalent • A statement formula A is said to be logically equivalent to a statement formula B if the statement formula A ↔ B is a tautology. If A is logically equivalent to B , then symbolically we write A ≡ B Discrete Mathematical Structures: Theory and Applications

  19. Mathematical Logic Discrete Mathematical Structures: Theory and Applications

  20. Mathematical Logic • Proof of (~p ^q ) → (~(q →p )) Discrete Mathematical Structures: Theory and Applications

  21. Mathematical Logic • Proof of (~p ^q ) → (~(q →p )) [continued] Discrete Mathematical Structures: Theory and Applications

  22. Validity of Arguments • Proof: an argument or a proof of a theorem consists of a finite sequence of statements ending in a conclusion • Argument: a finite sequence of statements. • The final statement, , is the conclusion, and the statements are the premises of the argument. • An argument is logically valid if the statement formula is a tautology. Discrete Mathematical Structures: Theory and Applications

  23. Validity of Arguments - Example Discrete Mathematical Structures: Theory and Applications

  24. Validity of Arguments • Valid Argument Forms • Modus Ponens (Method of Affirming) Discrete Mathematical Structures: Theory and Applications

  25. Validity of Arguments • Valid Argument Forms • Modus Tollens (Method of Denying) Discrete Mathematical Structures: Theory and Applications

  26. Validity of Arguments • Valid Argument Forms • Disjunctive Syllogisms • Disjunctive Syllogisms Discrete Mathematical Structures: Theory and Applications

  27. Validity of Arguments • Valid Argument Forms • Hypothetical Syllogism (proven earlier) • Dilemma Discrete Mathematical Structures: Theory and Applications

  28. Validity of Arguments • Valid Argument Forms • Conjunctive Simplification • Conjunctive Simplification Discrete Mathematical Structures: Theory and Applications

  29. Validity of Arguments • Valid Argument Forms • Disjunctive Addition • Disjunctive Addition Discrete Mathematical Structures: Theory and Applications

  30. Validity of Arguments • Valid Argument Forms • Conjunctive Addition Discrete Mathematical Structures: Theory and Applications

  31. Validity of Arguments – Formal Derivation • Prove • Formal Derivation Rule Comment1. P  Q Premise2. Q R Premise3. P Assumption Assume P4. Q 1,3, MP5. R 2,4, MP R is now proved6. P R DT Discharge P, ie, P is no longer to be used, and conclude that P  R • Uses Deduction Theorem (DT) Discrete Mathematical Structures: Theory and Applications

  32. Quantifiers and First Order Logic • Have dealt with Propositional Logic (Calculus) so far • Propositional variables, constants, expressions • Dealt with truth or falsity of expressions as a whole • Consider:1. All cats have tails2. Tom is a cat3. Tom has a tail • Cannot conclude 3, given 1 and 2 using propositional logic • Predicate Calculus – allows us to identify individuals such as Tom together with properties and predicates. Discrete Mathematical Structures: Theory and Applications

  33. Quantifiers and First Order Logic • Predicate or Propositional Function • Let x be a variable and D be a set; P(x) is a sentence • Then P(x) is called a predicate or propositional function with respect to the set D if for each value of x in D, P(x) is a statement; i.e., P(x) is true or false • Moreover, D is called the domain of the discourse and x is called the free variable Discrete Mathematical Structures: Theory and Applications

  34. Quantifiers and First Order LogicPropositional function example #1 Let P(x) be the statement: x is an odd integer Let D be the set of all positive integers. Then P is a propositional function with domain of discourse D. • For each x in D , P(x) is a proposition, i.e. a sentence which is either true or false. • P(1): 1 is an odd integer – True • P(14): 14 is an odd integer - False Discrete Mathematical Structures: Theory and Applications

  35. Quantifiers and First Order LogicPropositional function example #2 Let P(x) be the statement: the baseball player hit over .300 in 2003 Let D be the set of all baseball players. Then P is a propositional function with domain of discourse D. • For each x in D , P(x) is a proposition, i.e. a sentence which is either true or false. • P(Barry Bonds): Barry Bonds hit over .300 in 2003 - True • P(Alex Rodriguez): Alex Rodriguez hit over .300 in 2003 - False Discrete Mathematical Structures: Theory and Applications

  36. Quantifiers and First Order Logic • Predicate or Propositional Function • Example: • Q(x,y) : x > y, where the Domain is the set of integers • Q is a 2-place predicate • Q is T for Q(4,3) and Q is F for Q (3,4) Discrete Mathematical Structures: Theory and Applications

  37. Quantifiers and First Order Logic • Universal Quantifier • Let P(x) be a predicate and let D be the domain of the discourse. The universal quantification of P(x) is the statement: • For all x, P(x) or • For every x, P(x) • The symbol is read as “for all and every” • Two-place predicate: Discrete Mathematical Structures: Theory and Applications

  38. Quantifiers and First Order Logic • Universal Quantifier Examples • Consider the statement • It is true if P(x) is true for every x in D • It is false if P(x) is false for at least one x in D • Consider with D being the set of all real numbers. • The statement is true because for every real number x, it is true that the square of x is positive or zero. • Consider that with D being the set of real numbers is false. Why? Discrete Mathematical Structures: Theory and Applications

  39. Quantifiers and First Order Logic • Existential Quantifier • Let P(x) be a predicate and let D be the domain of the discourse. The existential quantification of P(x) is the statement: • There exists x, P(x) • The symbol is read as “there exists” • Bound Variable • The variable appearing in: or Discrete Mathematical Structures: Theory and Applications

  40. Quantifiers and First Order Logic • Existential Quantifier Example • Consider • It is true since there is at least one real number x for which the proposition is true. Try x=2 • Suppose that P is a propositional function whose domain of discourse consists of the elements d1,…,dn. The following pseudocode determines whether is true. Discrete Mathematical Structures: Theory and Applications

  41. Quantifiers and First Order Logic • Negation of Predicates (DeMorgan’s Laws) • Example: • If P(x) is the statement “x has won a race” where the domain of discourse is all runners, then the universal quantification of P(x) is , i.e., every runner has won a race. The negation of this statement is “it is not the case that every runner has won a race. Therefore there exists at least one runner who has not won a race. Therefore: and so, Discrete Mathematical Structures: Theory and Applications

  42. Quantifiers and First Order Logic • Negation of Predicates (DeMorgan’s Laws) Discrete Mathematical Structures: Theory and Applications

  43. Quantifiers and First Order Logic • Formulas in Predicate Logic • All statement formulas are considered formulas • Each n, n =1,2,...,n-place predicate P( ) containing the variables is a formula. • If A and B are formulas, then the expressions ~A, (A∧B), (A∨B) , A →B and A↔B are statement formulas, where ~, ∧, ∨, → and ↔ are logical connectives • If A is a formula and x is a variable, then ∀x A(x) and ∃x A(x) are formulas • All formulas constructed using only above rules are considered formulas in predicate logic Discrete Mathematical Structures: Theory and Applications

  44. Quantifiers and First Order Logic • Additional Rules of Inference • If the statement ∀x P(x) is assumed to be true, then P(a) is also true,where a is an arbitrary member of the domain of the discourse. This rule is called the universal specification (US) • If P(a) is true, where a is an arbitrary member of the domain of the discourse, then ∀x P(x) is true. This rule is called the universal generalization (UG) • If the statement ∃x P (x) is true, then P(a) is true, for some member of the domain of the discourse. This rule is called the existential specification (ES) • If P(a) is true for some member a of the domain of the discourse, then ∃x P(x) is also true. This rule is called the existential generalization (EG) Discrete Mathematical Structures: Theory and Applications

  45. Quantifiers and First Order Logic • Counterexample • An argument has the form ∀x (P(x ) → Q(x )), where the domain of discourse is D • To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true • This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of the above implication • To show that ∀x (P(x) → Q(x)) is false by finding an x in D such that P(x) → Q(x) is false is called the disproof of the given statement by counterexample Discrete Mathematical Structures: Theory and Applications

  46. Logic and CS • Logic is basis of ALU • Logic is crucial to IF statements • AND • OR • NOT • Implementation of quantifiers • Looping • Database Query Languages • Relational Algebra • Relational Calculus • SQL Discrete Mathematical Structures: Theory and Applications

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