Elementary Discrete Mathematics Jim Skon

1
Proofs

• Elementary Discrete MathematicsJim Skon

2
Proofs

• Why proofs?
• Careful examination to determine if mistake has
  been made.
• Convince someone else about proposition.

3
Proofs

• Proofs based on systems of rules.
• A set of rules should be
• consistent - can't prove anything invalid
• complete - can prove anything that is true.
• Problem Gödel has proved that any system of
  consistent rules is incomplete!

4
Proofs

• Proofs must be based on some underlying set of
  truths which, in general, everyone believes.
• Axioms or Postulates
• Definitions

5
Proofs - Axioms

• Axioms or Postulates - set of assumptions which
  are believed to be fundamentally true - no proof
  is given.
• Examples of Axioms
• Given two distinct points, there is exactly one
  line that contains them.
• for all real numbers xy yx

6
Proofs - Definitions

• Definitions - set of statements used to define
  new concepts in terms or existing ones. No proof
  needed.
• Examples of Definitions
• Two lines are parallel if they are on the same
  plain and never meet
• The absolute value x of a real number x is
  defined to be x if x is positive and -x
  otherwise.

7
Proofs - Definitions

• Example definitions
• x is even ax 2a
• x is odd ax 2a 1

8
Valid reasoning in proofs

• A mathematical proof is a sequence of statements,
  such that each statement
• 1. is an assumption, or
• 2. is a proposition already proved, or
• 3. Follow logically from one or more previous
  statements in proof.

9
Valid reasoning in proofs

• Logically follows
• A proposition Q follows logically from
  propositions P1, P2, ..., Pn if Q must
  be true whenever P1, P2, ..., Pn are
  true.

10
Valid reasoning in proofs

• Example modus ponens
• P Ù (P Q) Þ Q modus ponensP The car is
  running
• Q The car has gas.
• If we know that the car is running (P), we can
  prove that (Q) it has gas.
• Q Ù (P Q) Þ P non-logical implication

11
Rules of Inference

• Rules of Inference - used in proofs, or
  arguments, to move from what is known to what we
  want to prove.
• modus ponens is a valid rule of inference.

12
Argument

• Argument - consists of a collection of
  statements, called premises of the argument,
  followed by a conclusion statement.
• A1
• A2
• An
• ? A

Premises
Conclusion
13
Valid Argument

• An argument is said to be valid if whenever all
  the premises are true, the conclusion is also
  true.
• If the premises are true, but the conclusion
  false, the argument is said to be invalid.

14
Example (modus ponens)

• Prove
• If I have a cold, then I will not go to the game
• I have a cold
• Therefore, I will not go to the game
• p ?q p q p ?q
• p T T T
• ? q T F F
• F T T
• F F T

15
Example

• Prove
• If I do my homework, I will get a passing grade
  on the test.
• I passed the test.
• Therefore I did all my homework.
• p ?q
• q
• ? p
• Not Valid! A fallacy!
• Called the fallacy of affirming the conclusion.

16
Addition

• Prove
• It is windy outside.
• Therefore it is either windy outside or cloudy
  outside.
• p
• ? p ? q

17
Simplification

• Prove
• It is sunny and it is cold.
• Therefore it is sunny.
• p ? q
• ? p

18
Modus tollens

• Prove
• It is not cold today.
• If is was clear last night, then it will be cold
  today
• Therefore it was not clear last night.
• ?q
• p ?q
• ??p

19
Hypothetical syllogism

• Prove
• If Dr. Fairbanks is speaking in chapel today, I
  will not skip
• If I dont skip, I will have perfect attendance.
• Therefore If Dr. Fairbanks is speaking in chapel
  today, then I will have perfect attendance.
• p ? q
• q ? r
• ? p ? r

20
Disjuctive syllogism

• Prove
• I will work in the library today or I go fishing.
• I did not work in the library today
• Therefore I went fishing.
• p ? q
• ?p
• ? q

21
Hypothetical syllogism

• Prove
• If you love me you will keep my commandments.
  (Jn 1415)
• If you keep my commandments, you will abide in my
  love. (Jn 1510)
• Therefore, if you love me then you will abide in
  my love.
• This argument valid by the law of hypothetical
  syllogism.

22
Proofs

• Three Techniques
• Show true using logical inference
• Show true by showing that no way exists to make
  all premises true but conclusion false
• Show false by finding a way to make premises true
  but conclusion false.

23
Proof Example

• Consider
• ?q ? ?p
• q ? r
• p ? s
• ?r

24
Proof Example

• Consider
• p ? (q ? r)
• q
• ?p ? r
• Show no way to make all premises true but
  conclusion false

25
Proof by contradiction

• Consider
• r ? s
• p ??s
• r ? q
• ?p ? q
• Use proof by contradiction

26
Proof Example

• If the law is sufficient, then Christ died in
  vain
• The law is sufficient
• Therefore Christ died in vain.

27
Sorites

• Using proof by induction, we can show that the
  law of syllogism may be extended to more than two
  premises. This argument is called a sorites.
• p1 ? p2
• p2 ? p3
• pn-1 ? pn
• ?p1 ? pn

28
Sorites

• Romans 1013-15
• 1) Whoever will call upon the name of the Lord
  will be saved.
• 2) They must believe to call on the Lord.
• 3) They must hear the Gospel to believe.
• 4) They must have the word preached to them to
  hear the Gospel.
• 5) A person must be sent for the word to be
  preached.

29
Sorites

• p - He is saved
• q - He calls on the Lord
• r - He believes
• s - He hears the Gospel
• t - He has the word preached to him
• u - A person is send to preach

30
Sorites

• ?u ? ?t
• ?t ??s
• ?s ??r
• ?r ??q
• ?q ??p
• ??u ? ?p

If no one is sent to preach the gospel, then no
one will be saved!
31
Example

• Babies are illogical
• Nobody is despised who can manage a crocodile
• Illogical persons are despised
• Therefore, babies cannot manage crocodiles

32
Types of proof

• Vacuous Proof of P Q
• The truth value of P Q is true if P is false.
  If P can be shown false, then P Q holds.
• Thus prove P Q by showing P is false.

33
Trivial Proof of P Q

• If it is possible to establish that Q is true,
  then only the first and third lines of the truth
  table below apply.
• P Q P Q
• T T T
• T F F
• F T T
• F F T
• Thus prove P Q by showing Q true.

34
Direct Proof of P Q

• Prove Q, using P as an assumption.
• Thus prove P Q by showing Q is true whenever P
  is true.

35
Indirect Proof of P Q

• Prove the contrapositive, e.g. ØQ ØP is true,
  using one of the other proof methods.

36
Proof by contradiction

• Assume the negation of the proposition is true,
  then derive a contradiction.
• Thus to prove of P Q, assume P Ù ØQ is true,
  then derive a contradiction.

37
Proof by cases of P Q

• To prove P Q, find a set of propositions P1,
  P2, ..., Pn, n?2, in which at least one Pj must
  be true for P to be true. P P1? P2 ?
  ... ? Pn
• Then prove the n propositions P1 Q, P2
  Q, ..., Pn Q.

38
Vacuous Proof

• Consider the proposition
• If you your grandfather dies as a baby then you
  will get an A in this class.
• Proof of this statement
• Your grandfather didnt die, thus thus the
  premise must be false. Thus P Q must be true.

39
Trivial Proof

• Consider the proposition
• If 3n2 5n -2 ? 2n2 7n - 16 then n n2.
  P(n).
• Proof of P(0)
• 0 02, thus P(0) is trivially true. QED.

40
Direct Proof

• Consider The sum of two even numbers is even.
• Restate as
• "x"y (x is even and y is even) x y is even
• Proof
• 1. Remember x is even ax 2a (definition)
• 2. Assume x is even and y is even (assume
  hypothesis)
• 3. x y 2a 2b (from 1 and 2)
• 4. 2a 2b 2(ab)
• 5. By 1, 2(ab) is even - QED.

41
Direct Proof

• Consider Every multiple of 6 is also a multiple
  of 3.
• Rewrite "x zy(6x y 3z y)
• Proof
• 1. Assume 6x y (hypothesis)
• 2. 6x y can be rewritten as 3 2x y
• 3. Let z 2x, then 3z y holds. QED.

42
Indirect Proofs

• Prove the contrapositive, e.g. Prove that
• ØQ ØP
• is true

43
Indirect Proofs

• Prove If x2 is even, then x is even.
• Rewrite "x (EVEN(x2) EVEN(x))

44
Indirect Proofs

• Prove If x2 is even, then x is even
• 1. "x (ODD(x) ODD(x2)) (contrapositive)
• 2. Assume 1 ODD(n) true for some n (hypothesis)
• 3. x is odd ax 2a 1 (definition)
• 4. n 2a 1 for some a (2 3)
• 5. n2 (2a 1)2 (substitution)
• 6. (2a 1)2 (2a 1)(2a 1)
• 4a 2 4a 1
• 2 (2a2 2a) 1
• 7. 2 (2a2 2a) 1 is odd (3 6) QED

45
Proof by contradiction

• To prove of P Q, assume Ø(P Q), derive a
  contradiction.
• Recall that P Q ? ØP ? Q
• Then Ø(P Q) ? Ø(ØP ? Q) ?
  P Ù ØQ (Demorgans)
• Thus to prove P Q we assume P Ù ØQ and show a
  contradiction.

46
Proof by contradiction

• Consider Theorem There is no largest prime
  number.
• This can be stated as
• "If x is a prime number, then there exists
  another prime y which is greater"
• Formally "x y (PRIME(x) Ù PRIME(y) x lt
  y)

47
Proof by contradiction

• There is no largest prime number
• Assume largest prime number does exist. Call
  this number p.
• Restate implication as p is prime, and there
  does not exist a prime which is greater.
• 1. Form a product r 2 3 5 ... p)
• (e.g. r is the product of all primes)
• 2. If we divide r1 by any prime, it will have
  remainder 1
• 3. r1 is prime, since any number not divisible
  by any prime which is less must be prime.
• 4. but r1 gt p , which contradicts that p is the
  greatest prime number. QED.

48
Proof by cases

• To prove P Q, find a set of propositions P1,
  P2, ..., Pn, n?2, in which at least one Pj must
  be true for P to be true. P P1? P2 ?
  ... ? PnThen prove the n propositions
  P1 Q, P2 Q, ..., Pn Q.
• ThusP(P1ÚP2Ú...ÚPn) and (P1Q)Ù(P2Q)Ù...Ù(PnQ
  )Þ(PQ)

49
Proof by cases

• Consider For every nonzero integer x ,x2 gt 0.
• LetP "x is a nonzero integerQ x2 gt 0
• We want to prove P Q

50
Proof by cases

• If P "x is a nonzero integer Q x2 gt 0
• Prove P Q
• P can be broken up into two cases
• P1 x gt 0
• P2 x lt 0
• Note that P (P1 Ú P2).

51
Proof by cases

• For every nonzero integer x ,x2 gt 0.
• Prove each case -
• Prove P1 Q
• If x gt 0, then x2 gt 0, since the product of
  two positive numbers is always positive.
• Prove P2 QIf x lt 0, then x2 gt 0, since the
  product of two negative numbers is always
  positive. QED.