Truth and Proof Math vs. Reality Propositions & Predicates Mathematics for Computer Science MIT 6.042J/18.062J Truth and Proof Math vs. Reality Propositions & Predicates Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal

Only Prime Numbers? Let . Hypothesis: is a prime number

. . . . . . Only Prime Numbers? Evidence: prime prime prime prime prime looking good! . . . prime enough already!

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

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

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

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

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

Further Extreme Example Hypothesis: has no natural number solution.

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

MATHEMATICIAN: 3 is prime, 5 is prime, 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!

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!

LAWYER: Ladies and Gentleman of the 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.

Math Sets Numbers Booleans Strings Functions Relations Vectors

Not Math Solar System

Not Math Physical Motion

Not Math Family

Not Math Cats

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

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

Propositional (Boolean) Logic Proposition is either True or False

Propositional (Boolean) Logic Proposition is either True or False Example:

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

Operators (if and only if)

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.''

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.''

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

Could use a truth table. Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. T T T T T F T F T T F F F T T F T F F F T F F F

Conclusion (below the line) must be true whenever Truth Table Conclusion (below the line) must be true whenever Hypotheses (above the line) are true. T T T T T T F F T F T F T F F F F T T F F T F F F F T F F F F T

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

Every even integer greater than 2 is the sum of two primes. Goldbach Conjecture Every even integer greater than 2 is the sum of two primes.

. . . Every even integer greater than 2 is the sum of two primes. Goldbach Conjecture Every even integer greater than 2 is the sum of two primes. Evidence: . . .

. . . Every even integer greater than 2 is the sum of two primes. Goldbach Conjecture Every even integer greater than 2 is the sum of two primes. Evidence: . . .

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

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

Goldbach Conjecture The answer is on my desk!

Goldbach Conjecture The answer is on my desk! (Proof by Cases)

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.

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

Tutorial Problems 1 & 2

Predicates Predicates are Propositions with variables: Example:

Predicates For x = 1 and y = 3, equation is true: is true For x = 1 and y = 4, equation is false: is false

Quantifiers For ALL x There EXISTS some y

Quantifiers For ALL x There EXISTS some y x, y range over Domain of Discourse True over Domain

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

Quantifiers True over positive real numbers, False over negative real numbers,

Validity True no matter what predicate Q is, the Domain is.

Tutorial Exercises 3-5