1. 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

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

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

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

5. 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.

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

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

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

9. Further Extreme Example
Hypothesis:
has no natural number solution.

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

11. 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!

12. 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!

13. 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.

14. Math
Sets
Numbers
Booleans
Strings
Functions
Relations
Vectors

15. Not Math
Solar System

16. Not Math
Physical Motion

17. Not Math
Family

18. Not Math
Cats

19. René Descartes' MEDITATIONS
Cogito ergo sum
René Descartes' MEDITATIONS
(Picture source:

20. 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.

21. Propositional (Boolean) Logic
Proposition is either True or False

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

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

24. Operators
(if and only if)

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.''

26. 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.''

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

28. 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

29. 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

30. 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!

31. 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.

32. . . . 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: . . .

33. . . . 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: . . .

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

35. 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!…

36. Goldbach Conjecture
The answer is on my desk!

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

38. 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.

39. 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

40. Tutorial Problems 1 & 2

41. Predicates
Predicates are Propositions with variables:
Example:

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

43. Quantifiers
For ALL x
There EXISTS some y

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

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

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

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

48. Tutorial Exercises 3-5