Lecture 2: Foundations: Mathematical Logic

1. Lecture 2: Foundations: Mathematical Logic Discrete Mathematical Structures: Theory and Applications

2. Learning Objectives • Learn about sets • Explore various operations on sets • Become familiar with Venn diagrams • Learn how to represent sets in computer memory • Learn about statements (propositions) Discrete Mathematical Structures: Theory and Applications

3. Learning Objectives • Learn how to use logical connectives to combine statements • Explore how to draw conclusions using various argument forms • Become familiar with quantifiers and predicates • Learn various proof techniques • Explore what an algorithm is Discrete Mathematical Structures: Theory and Applications

4. 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

5. 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

6. 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 Discrete Mathematical Structures: Theory and Applications

7. 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

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

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

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

11. 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

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

13. 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

14. 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

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

16. Mathematical Logic • Statement Formulas • Definitions • Symbols p ,q ,r ,...,called statement variables • Symbols ~, ∧, ∨, →,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 ∨ 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

17. Mathematical Logic • Precedence of logical connectives is: • ~ highest • ∧ second highest • ∨ third highest • → fourth highest • ↔ fifth highest Discrete Mathematical Structures: Theory and Applications

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

19. 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

20. 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 (or A ⇔ B) Discrete Mathematical Structures: Theory and Applications

21. Mathematical Logic Discrete Mathematical Structures: Theory and Applications

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

23. Mathematical Logic • Proof of (~p ∧q ) → (~(q →p )) [Continued] Discrete Mathematical Structures: Theory and Applications

24. 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

25. Validity of Arguments • Valid Argument Forms • Modus Ponens (Method of Affirming) • 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 • 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