Lecture 8 Introduction to Logic

1. Lecture 8Introduction to Logic CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine

2. Lecture Introduction • Reading • Kolman - Section 2.1 • Logical Statements • Logical Connectives / Compound Statements • Negation • Conjunction • Disjunction • Truth Tables • Quantifiers • Universal • Existential CSCI 1900

3. Statement of Proposition • Proposition (statement of proposition) – a declarative sentence that is either true or false, but not both • Examples: • Star Trek was a TV series: True • 2+3 = 5: True • Do you speak Klingon? This is a question, not a statement CSCI 1900

4. Statement of Proposition (cont) • x - 3 = 5: a declarative sentence, but not a statement since it is true or false depending on the value of x • Take two aspirins: a command, not a statement • The current temperature on the surface of the planet Venus is 800oF: a proposition of whose truth is unknown to us • The sun will come out tomorrow: a proposition that is either true or false, but not both, although we will have to wait until tomorrow to determine the answer CSCI 1900

5. Notation • x, y, z, … denote variables that can represent real numbers • p, q, r,… denote propositional variables that can be replaced by statements • p: The sun is shining today • q: It is cold CSCI 1900

6. Negation (Not) • If p is a statement, the negation of p is the statement not p • Denoted ~p ( alternately: !p or p or p ) • If p is True, • ~p is False • If p is False, • ~p is True • not is a unary operator for the collection of statements and ~p is a statement if p is a statement CSCI 1900

7. Negation Truth Table CSCI 1900

8. Examples of Negation • If p: 2+3 > 1 Then ~p: 2+3 < 1 • If q: It is night Then • ~q: It is not the case that it is night, • It is not night • It is day CSCI 1900

9. Conjunction • If p and q are statements • The conjunction of p and q is the compound statement “p and q” • Denoted p  q • p  q is true only if both p and q are true • Example: • p: ETSU parking permits are readily available • q: ETSU has plenty of parking • p  q = ? CSCI 1900

10. Conjunction Truth Table CSCI 1900

11. English Conjunctives • The following are common conjunctives in English: And, Now, But, Still, So, Only, Therefore, Moreover, Besides, Consequently, Nevertheless, For, However, Hence, Both... And, Not only... but also, While, Then, So then CSCI 1900

12. Disjunction (Inclusive) • If p and q are statements, • The (inclusive) disjunction of p and q is the compound statement “p or q” • Denoted p  q • p  q is true if either p is true or q is true or both are true • Example: • p: I am a male q: I am under 90 years old • p  q = ? • p: I am a male q: I am under 20 years old • p  q = ? CSCI 1900

13. Disjunction (Inclusive) Truth Table CSCI 1900

14. Exclusive Disjunction • If p and q are statements • The exclusive disjunction true if either p is true or q is true, but not both are true • Denoted p  q • Example: • p: It is daytime • q: It is night time • p  q (in the exclusive sense) = ? CSCI 1900

15. Disjunction(Exclusive) Truth Table CSCI 1900

16. Exclusive versus Inclusive • Depending on the circumstances, some English disjunctions are inclusive and some of exclusive. • Examples of Inclusive • “I have a dog” or “I have a cat” • “It is warm outside” or “It is raining” • Examples of Exclusive • Today is either Tuesday or it is Thursday • The light is either on or off CSCI 1900

17. Compound Statements • A compound statement is a statement made by joining simple propositions with logical connectors • For n individual propositions, there are 2n possible combinations of truth values • The truth table contains 2n rows identifying the truth values for the statement represented by the table • Use parenthesis to denote order of precedence •  has precedence over  • Use parenthesis to ensure meaning is clear CSCI 1900

18. Truth Tables as Tools • Compound statements can be easily and systematically investigated with truth tables • Assign a portion of the compound statement to a column • Final column represents the complete compound statement CSCI 1900

19. Compound Statement Example (pq) (~p) CSCI 1900

20. Quantifiers • Recall from Section 1.1, a set may be defined by its properties {x | P(x)} • For a specific element e to be a member of the set, P(e) must evaluate to “true” • P(x) is called a predicate or a propositional function CSCI 1900

21. Computer Science Functions • if P(x) then execute certain steps • while Q(x) do specified actions CSCI 1900

22. Universal Quantification • Universal quantification of the predicate P(x) means “For all values of x, P(x) is true” • Denoted x P(x) • The symbol  is called the universal quantifier • The order in which multiple universal quantifications are applied does not matter (e.g., xy P(x,y) ≡ yx P(x,y) ) CSCI 1900

23. Universal Examples: • P(x): -(-x) = x • This predicate makes sense for all real numbers x R • The universal quantification of P(x), x P(x), is a true statement, because for all real numbers, -(-x) = x • Q(x): x+1<4 • x Q(x) is a false statement, because, for example, Q(5) is not true CSCI 1900

24. Existential Quantification • Existential quantification of a predicate P(x) is the statement: “There exists a value of x for which P(x) is true.” • Denoted x P(x) • Existential quantification may be applied to several variables in a predicate • The order in which multiple existential quantifications are considered does not matter CSCI 1900

25. Existential Examples: • P(x): -(-x) = x • The existential quantification of P(x), x P(x), is a true statement, because there is at least one real number where -(-x) = x • Q(x): x+1<4 •  x Q(x) is a true statement, because, for example, Q(2) is true • Nota Bene:  is not the complement of  • An Example • N(x): x≠x • x N(x) is false and x N(x) is also false CSCI 1900

26. Applying Both  and  Quantifications • Order of application matters • Example:Let A and B be n x n matrices • The statement A ( B | A + B = In ) • Reads: for every A there is a B such that A + B = In • Prove by coming up for equations for bii and bij (ji) • Now reverse the order: B (A |A + B = In) • Reads: there exists a B, for all A, such that ,A + B = In • The second proposition is F A L S E ! CSCI 1900

27. Assigning Quantification - 1 • Let p: x R(x) [R(x): a person is good] • If p is true then  x ~R(x) is false • If every person is good, then there does not exist person who is not good • If p is true then  x R(x) is true • If every person is good, then there exists a person who is good • If p is false then  x ~R(x) is true • If not every person is good, then there exists a person who is not good CSCI 1900

28. Assigning Quantification - 2 • Let p: x R(x) [R(x): a person is good] • If p is true then x ~R(x) is false • If there exists a person who is good, then it is not true that all people are not good • If p is false then x ~R(x) is true • If there does not exist a good person then every person is not good CSCI 1900

29. Implications of the Previous Slide • Assume a statement is made that “for all x, P(x) is true” • If we can find one case that is not true • The statement is false • If there isn’t one case that is not true • The statement is true • Example:  positive integers, n, n P(n) = n2- n+ 41 is a prime number • This is false because  an integer resulting in a non-prime value, i.e.,  n such that P(n) is false CSCI 1900

30. Key Concepts Summary • Statement of Proposition • Logical Connectives / Compound Statements • Negation • Conjunction • Disjunction • Truth Tables • Quantifiers • Universal • Existential CSCI 1900