CSC165. Larry Zhang, October 7, 2014

Transcription

1 CSC165 Larry Zhang, October 7, 2014

2 If you did bad, then it is not bad. Proof: assume you left all questions blank # that s pretty bad! then you get 20% # rule on test paper assume class average is 70% # pretty high! then you are 50% below average in this term test # 70% - 20% = 50% then this term test weighs 6% of final grade # according course info sheet then you are 3% below average in term of final grade # 6%x50%=3% then it is below are the acceptable margin of error # 5% in physics then it is totally acceptable then it is not bad even if you left everything blank and others did well!

3 Today s outline proof using contrapositive proof using contradiction proof for existence proof about a sequence

4 Lecture 5.1 contrapositive, contradiction Course Notes: Chapter 3

5 Last week direct proof for universally quantified implication as example, we proved

6 The proof write comments to explain each step # n is a generic natural number # definition of odd # square of both sides # some algebra # some algebra # definition of odd # introduce => # introduce

7 Now, we want to prove... not odd not odd is the contrapositive of what we knew The proof for P => Q also proves Q => P

8 The proof # n is a generic natural number # definition of odd... same proof for n odd => n² odd # definition of odd # introduce => # contrapositive # introduce

9 Proof using contrapositive General idea: Instead of proving P => Q, prove Q => P (equivalent to P => Q) Chain of implication: When is this useful? when the reverse direction is easier to prove than the original.

10 sometimes the reverse direction is easier to prove

11 Try the original direction P => Q # definition of odd stuck!?

12 Try the contrapositive Q => P # definition of even # algebra easy! # definition of even

13 takeaway When it s not easy to prove P => Q, try proving Q => P

14 contradiction a special case of contrapositive

15 P => Q, with implicit P sometimes it is not clear what P is, for example, There are infinitely many even natural numbers. Q If every common sense is true, then Q contrapositively... De Morgan s if not Q, then something in common sense is false if not Q, then it will contradict with some common sense

16 thoughts There are infinitely many even natural numbers. Suppose there is a finite number of even numbers then there must be a largest one, call it X... but if I double X I get a larger number, a larger even number so X is NOT the largest one Contradiction!

17 There are infinitely many even natural numbers. Proof: assume there are a finite number of even numbers # assuming Q then there exists a largest even number, m then let m = 2m then m is an even number, and larger than m # contradiction, since m is the largest even number then there are infinitely many even numbers # assuming Q leads to contradiction, so Q.

18 There are 5 boxes in which there are in total 51 balls. Prove that there is a box with at least 11 balls in it. Proof: assume there is not a box with at least 11 balls # not Q then all boxes have at most 10 balls # ( x, P(x)) x, P(x) then 5 boxes have at most 50 balls # contradiction with that there are in total 51 balls then there is a box with at least 11 balls. # Q is true

19 getting ready for a more challenging one...

20 Prime number A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers: 2, 3, 5, 7, 11,, 997, Not prime numbers: 0, 1, 4, 6, 8,, 1000,... except 0 and 1, all non-primes ( >2 ) can be divided by a prime.

21 There are infinitely many prime numbers.

22 thoughts There are infinitely many prime numbers. Suppose there are finitely many prime numbers feels like a prime, but can t just say that Alright, suppose it s not prime... then it can be divided by a prime absurd!

23 Proof: # negation of S # q is the product of all primes # q is at least 2 x 3 = 6 # q is the product of all primes # px divides both q+1 and q # every integer > 2 has a prime divisor # only 1 divides 1 # contradiction! 1 is not a prime number! # assume S leads to contradiction

24 takeaway When the assumptions are implicit, try assuming Q, and see whether it leads somewhere, hunting for a contradiction.

25 Lecture 5.2 existential, sequence Course Notes: Chapter 3

26 direct proof of the existential How to prove: find a single example.

27 Prove: There exists a car that is red. Proof: pick a car x then x is red # by human visual checking then there exists a car that is red # introduce

28 thoughts just find a valid x!

29 Proof: # choose an example # well known fact: 2 is real # plug in x = 2 # example given

30 prove a claim about a sequence

31

32 thoughts j aj Pick i = 1 not good, counter-example: j=1?

33 thoughts j aj Pick i = 2 looks good!?

34 thoughts j aj Pick i = 4? Also good! Ok, let s use it.

35 Proof # a generic element of N # negate consequent for contrapositive # i = 4 # i = 4 # def of sequence # introduce assumption # contrapositive, knowing j derive aj # introduce for all # introduce exists

36 takeaway one proof may consist of several different structures with multiple quantifiers, zoom into smaller and smaller part of the statement, layer by layer

37 next week more proofs...