Mathematics for Computer Science MIT 6 042 J18

Mathematics for Computer Science MIT 6. 042 J/18. 062 J Truth and Proof Math vs. Reality Propositions & Predicates Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal February 7, 2002 1

Only Prime Numbers? Let . Hypothesis: is a prime number February 7, 2002 2

Only Prime Numbers? Evidence: prime . . prime looking good! prime enough already! February 7, 2002 3

Only Prime Numbers? Let . is a prime number This can’t be a coincidence. The hypothesis must be true. February 7, 2002 4

Only Prime Numbers? Let . is a prime number This can’t be a coincidence. The hypothesis must be true. BUT NOT TRUE: is NOT PRIME. February 7, 2002 5

Only Prime Numbers? EXERCISE: Prove that 1601 is prime, and 1681 is not prime. February 7, 2002 6

Further Extreme Example EULER'S CONJECTURE (1769) has no solution when are positive integers: February 7, 2002 7

Further Extreme Example EULER'S CONJECTURE (1769) Counterexample: 218 years later by Noam Elkies at Liberal Arts school up Mass Ave: February 7, 2002 8

Further Extreme Example Hypothesis: has no natural number solution. February 7, 2002 9

Further Extreme Example Hypothesis: has no natural number solution. False. But smallest counterexample has MORE THAN 1000 digits! February 7, 2002 10

Evidence vs. Proof Claim: All odd numbers greater than 1 are prime. MATHEMATICIAN: 3 is prime, 5 is prime, 7 is prime, but is not prime, so the proposition is false! February 7, 2002 11

Evidence vs. Proof Claim: All odd numbers greater than 1 are prime. PHYSICIST: 3 is prime, 5 is prime, 7 is prime, 9 is not prime, but 11 is prime, 13 is prime. So 9 must be experimental error; the proposition is true! February 7, 2002 12

Evidence vs. Proof Claim: All odd numbers greater than 1 are prime. LAWYER: Ladies and Gentleman of the jury, it is beyond all reasonable doubt that odd numbers are prime. The evidence is clear: 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime, and so on. February 7, 2002 13

Math Sets Numbers Booleans Strings Functions Relations Vectors February 7, 2002 14

Not Math Solar System February 7, 2002 15

Not Math Physical Motion February 7, 2002 16

Not Math Family February 7, 2002 17

Not Math Cats February 7, 2002 18

Cogito ergo sum René Descartes' MEDITATIONS (Picture source: http: //www. btinternet. com/~glynhughes/squashed/descartes. htm ) February 7, 2002 19

Cogito ergo sum René Descartes' MEDITATIONS on First Philosophy in which the Existence of God and the Distinction Between Mind and Body are Demonstrated. February 7, 2002 20

Propositional (Boolean) Logic Proposition is either True or False February 7, 2002 21

Propositional (Boolean) Logic Proposition is either True or False Example: February 7, 2002 22

Propositional (Boolean) Logic Proposition is either True or False Example: Nonexamples: Wake up! Where am I? February 7, 2002 23

Operators (if and only if) February 7, 2002 24

Deductions A student is trying to prove that propositions P, Q, and R are all true. She proceeds as follows. First, she proves three facts: P implies Q, Q implies R, and R implies P. Then she concludes, ``Thus obviously P, Q, and R are all true. '' February 7, 2002 25

Deductions A student is trying to prove that propositions P, Q, and R are all true. She proceeds as follows. First, she proves three facts: P implies Q, Q implies R, and R implies P. Then she concludes, ``Thus obviously P, Q, and R are all true. '' February 7, 2002 26

Truth Table Could use a truth table. Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. February 7, 2002 27

Truth Table Could use a truth table. Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. T T F F T F T F February 7, 2002 28

Truth Table Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. T T F F T F T F T F F F T February 7, 2002 29

Truth Table Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. T F F F T T F F F F OK OK NOT OK! February 7, 2002 30

Goldbach Conjecture Every even integer greater than 2 is the sum of two primes. February 7, 2002 31

Goldbach Conjecture Every even integer greater than 2 is the sum of two primes. Evidence: . . February 7, 2002 32

Goldbach Conjecture Every even integer greater than 2 is the sum of two primes. Evidence: . . February 7, 2002 33

Goldbach Conjecture True for all even numbers with up to 13 digits! (Rosen, p. 182) It remains an OPEN problem: no counterexample, no proof. February 7, 2002 34

Goldbach Conjecture True for all even numbers with up to 13 digits! (Rosen, p. 182) It remains an OPEN problem: no counterexample, no proof. UNTIL NOW!… February 7, 2002 35

Goldbach Conjecture The answer is on my desk! February 7, 2002 36

Goldbach Conjecture The answer is on my desk! (Proof by Cases) February 7, 2002 37

Quicker by Cases Case 1: P is true. If the Hypothesis is true, then q must be true (because p implies q). Then r must be true (because q implies r). So the conclusion is true OK. February 7, 2002 38

Quicker by Cases Case 2: P is false. If the Hypothesis is true, then q must be false (because p implies q). Then r must be false (because q implies r). So the conclusion is (very) False February 7, 2002 39

Tutorial Problems 1 & 2 February 7, 2002 40

Predicates are Propositions with variables: Example: February 7, 2002 41

Predicates For x = 1 and y = 3, equation is true: is true For x = 1 and y = 4, equation is false: is false is true February 7, 2002 42

Quantifiers For ALL x There EXISTS some y February 7, 2002 43

Quantifiers For ALL x There EXISTS some y x, y range over Domain of Discourse True over Domain February 7, 2002 44

Quantifiers For ALL x There EXISTS some y x, y range over Domain of Discourse True over Domain False over Domain February 7, 2002 45

Quantifiers True over positive real numbers, False over negative real numbers, February 7, 2002 46

Validity True no matter what • predicate Q is, • the Domain is. February 7, 2002 47

Tutorial Exercises 3 -5 February 7, 2002 48