CS322

1. Week 4 - Friday CS322

2. Last time • What did we talk about last time? • We had a snow day • But you should have read about: • Proof by cases • Floor and ceiling • Indirect proofs

3. The Law of Small Numbers • I have claimed that many things can be demonstrated for a small set of numbers that are not actually true for all numbers • Example: • GCD(x,y) gives the greatest common divisor of x and y • GCD(n17 + 9, (n+1)17 + 9) = 1 for all n < 8424432925592889329288197322308900672459420460792433, but not for that number

4. Questions?

5. Logical warmup • Two friends who live 36 miles apart decide to meet and start riding their bikes towards each other. • They plan to meet halfway. • Each is riding at 6mph. • One of them has a pet carrier pigeon who starts flying the instant the friends start traveling. • The pigeon flies back and forth at 18mph between the friends until the friends meet. • How many miles does the pigeon travel?

6. Student Lecture Proof by Contradiction

7. Indirect Proof

8. Proof by contradiction • The most common form of indirect proof is a proof by contradiction • In such a proof, you begin by assuming the negation of the conclusion • Then, you show that doing so leads to a logical impossibility • Thus, the assumption must be false and the conclusion true

9. Example • Theorem: x, y  Z+, x2 – y2  1 • Proof by contradiction: Assume there is such a pair of integers

10. Two Classic Results

11. Square root of 2 is irrational Theorem: is irrational Proof by contradiction: • Suppose is rational • = m/n, where m,nZ, n  0 and m and n have no common factors • 2 = m2/n2 • 2n2 = m2 • 2k = m2, kZ • m = 2a, aZ • 2n2 = (2a)2 = 4a2 • n2 = 2a2 • n = 2b, bZ • 2|m and 2|n • is irrational QED • Negation of conclusion • Definition of rational • Squaring both sides • Transitivity • Square of integer is integer • Even x2 implies even x (Proof on p. 202) • Substitution • Transitivity • Even x2 implies even x • Conjunction of 6 and 9, contradiction • By contradiction in 10, supposition is false

12. Proposition 4.7.3 • Claim: • Proof by contradiction: Suppose such that 1 1 Contradiction Negation of conclusion Definition of divides Definition of divides Subtraction Substitution Distributive law Definition of divides Since 1 and -1 are the only integers that divide 1 Definition of prime Statement 8 and statement 9 are negations of each other By contradiction at statement 10 QED

13. Infinitude of primes Theorem: There are an infinite number of primes Proof by contradiction: • Suppose there is a finite list of all primes: p1, p2, p3, …, pn • Let N = p1p2p3…pn + 1, N Z • pk | N where pkis a prime • pk | p1p2p3…pn + 1 • p1p2p3…pn = pk(p1p2p3…pk-1pk+1…pn) • p1p2p3…pn = pkP, P Z • pk | p1p2p3…pn • pk does not divide p1p2p3…pn + 1 • pk does and does not divide p1p2p3…pn + 1 • There are an infinite number of primes QED • Negation of conclusion • Product and sum of integers is an integer • Theorem 4.3.4, p. 174 • Substitution • Commutativity • Product of integers is integer • Definition of divides • Proposition from last slide • Conjunction of 4 and 8, contradiction • By contradiction in 9, supposition is false

14. A few notes about indirect proof • Don't combine direct proofs and indirect proofs • You're either looking for a contradiction or not • Proving the contrapositive directly is equivalent to a proof by contradiction

15. Propositional Logic Review

16. Propositional logic • Statements • AND, OR, NOT, IMPLIES • Truth tables • Logical equivalence • De Morgan's laws • Tautologies and contradictions

17. Laws of Boolean algebra

18. Implications • Can be used to write an if-then statement • Contrapositive is logically equivalent • Inverse and converse are not (though they are logically equivalent to each other) • Biconditional: • p q  q  p

19. Arguments • A series of premises and a conclusion • Using the premises and rules of inference, an argument is valid if and only if you can show the conclusion • Rules of inference: • Modus Ponens • Modus Tollens • Generalization • Specialization • Conjunction • Elimination • Transitivity • Division into cases • Contradiction rule

20. Digital logic • The following gates have the same function as the logical operators with the same names: • NOT gate: • AND gate: • OR gate:

21. Predicate Logic Review

22. Predicates • A predicate is a sentence with a fixed number of variables that becomes a statement when specific values are substituted for to the variables • The domain gives all the possible values that can be substituted • The truth set of a predicate P(x) are those elements of the domain that make P(x) true when they are substituted

23. Sets • We will frequently be referring to various sets of numbers in this class • Some typical notation used for these sets: • Some authors use Z+ to refer to non-negative integers and only N for the natural numbers

24. Quantifiers • The universal quantifier  means “for all” • The statement “All DJ’s are mad ill” can be written more formally as: • x  D, M(x) • Where D is the set of DJ’s and M(x) denotes that x is mad ill • The existential quantifier  means “there exists” • The statement “Some emcee can bust a rhyme” can be written more formally as: • y  E, B(y) • Where E is the set of emcees and B(y) denotes that y can bust a rhyme

25. Negating quantified statements • When doing a negation, negate the predicate and change the universal quantifier to existential or vice versa • Formally: • ~(x, P(x))  x, ~P(x) • ~(x, P(x))  x, ~P(x) • Thus, the negation of "Every dragon breathes fire" is "There is one dragon that does not breathe fire"

26. Vacuously true • Any statement with a universal quantifier whose domain is the empty set is vacuously true • When we talk about "all things" and there's nothing there, we can say anything we want • "All mythological creatures are real." • Every single one of the (of which there are none) is real

27. Conditionals • Recall: • Statement: p q • Contrapositive: ~q  ~p • Converse: q  p • Inverse: ~p  ~q • These can be extended to universal statements: • Statement: x, P(x)  Q(x) • Contrapositive: x, ~Q(x)  ~P(x) • Converse: x, Q(x)  P(x) • Inverse: x, ~P(x)  ~Q(x) • Similar properties relating a statement equating a statement to its contrapositive (but not to its converse and inverse) apply

28. Necessary and sufficient • p is a sufficient condition for q means pq • p is a necessary condition for q means qp • These come over into universal conditional statements as well: • x, P(x) is a sufficient condition for Q(x) means x, P(x) Q(x) • x, P(x) is a necessary condition for Q(x) means x, Q(x) P(x)

29. Multiple quantifiers • With multiple quantifiers, we imagine that corresponding “actions” happen in the same order as the quantifiers • The action for x  A is something like, “pick any x from A you want” • Since a “for all” must work on everything, it doesn’t matter which you pick • The action for y  B is something like, “find some y from B” • Since a “there exists” only needs one to work, you should try to find the one that matches

30. Negating or changing multiple quantifiers • For negation, • Simply switch every  to  and every  to  • Then negate the predicate • Changing the order of quantifiers can change the truth of the whole statement but does not always • Furthermore, quantifiers of the same type are commutative: • You can reorder a sequence of  quantifiers however you want • The same goes for  • Once they start overlapping, however, you can’t be sure anymore

31. Quantification in arguments • Universal instantiation: If a property is true for everything in a domain (universal quantifier), it is true for any specific thing in the domain • Universal modus ponens: • x, P(x)  Q(x) • P(a) for some particular a • Q(a) • Universal modus tollens: • x, P(x)  Q(x) • ~Q(a) for some particular a • ~P(a)

32. Proofs Review

33. Proving existential statements and disproving universal ones • To prove x  D  P(x) we need to find at least one element of D that makes P(x) true • To disprove x  D, P(x)  Q(x), we need to find an x that makes P(x) true and Q(x) false

34. Proving universal statements • If the domain is finite, we can use the method of exhaustion, by simply trying every element • Otherwise, we can use a direct proof • Express the statement to be proved in the form x  D,if P(x) then Q(x) • Suppose that x is some specific (but arbitrarily chosen) element of D for which P(x) is true • Show that the conclusion Q(x) is true by using definitions, other theorems, and the rules for logical inference • Direct proofs should start with the word Proof, end with the word QED, and have a justification next to every step in the argument • For proofs with cases, number each case clearly and show that you have proved the conclusion for all possible cases

35. Definitions • If n is an integer, then: • n is even   k  Z  n = 2k • n is odd   k  Z  n = 2k + 1 • If n is an integer where n > 1, then: • n is prime   r  Z+,  s  Z+, if n = rs, then r = 1 or s = 1 • n is composite   r  Z+,  s  Z+  n = rs and r  1 and s  1 • r is rational   a, b Z  r = a/b and b  0 • For n, d  Z, • n is divisible by d  k Z  n = dk • For any real number x, the floor of x, written x, is defined as follows: • x = the unique integer n such that n ≤ x < n + 1 • For any real number x, the ceiling of x, written x, is defined as follows: • x = the unique integer n such that n – 1 < x ≤ n

36. Theorems • Unique factorization theorem: For any integer n > 1, there exist a positive integer k, distinct prime numbers p1, p2, …, pk, and positive integers e1, e2, …, ek such that • Quotient remainder theorem: For any integer n and any positive integer d, there exist unique integers q and r such that • n = dq + r and 0 ≤ r < d

37. A proof by cases • Theorem: for all integers n, 3n2 + n + 14 is even • How could we prove this using cases? • Be careful with formatting

38. Quiz

39. Upcoming

40. Next time… • Exam 1!

41. Reminders • Exam 1 is Monday in class!