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Elementary Discrete Mathematics Jim Skon

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Title: 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.
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