Proofs1 Elementary Discrete Mathematics Jim Skon.
-
Upload
alice-henry -
Category
Documents
-
view
223 -
download
0
Transcript of Proofs1 Elementary Discrete Mathematics Jim Skon.
Proofs 1
ProofsElementary Discrete Mathematics
Jim Skon
Proofs 2
Proofs
Why proofs? Careful examination to determine if mistake
has been made. Convince someone else about proposition.
Proofs 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!
Proofs 4
Proofs
Proofs must be based on some underlying set of truths which, in general, everyone believes.
Axioms or PostulatesDefinitions
Proofs 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
Proofs 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.
Proofs 7
Proofs - Definitions
Example definitions:
x is even a:x = 2a
x is odd a:x = 2a + 1
Proofs 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.
Proofs 9
Valid reasoning in proofsLogically follows:
A proposition Q follows logically from propositions P1, P2, ..., Pn if Q must be true whenever P1, P2, ..., Pn are true.
Proofs 10
Valid reasoning in proofs
Example: modus ponens P (P Q) Q modus ponens
P: 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 (P Q) P non-logical implication
Proofs 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.
Proofs 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
Proofs 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.
Proofs 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 qp T T T q T F FF T TF F T
Proofs 15
ExampleProve:
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.
Proofs 16
Addition
Prove: It is windy outside. Therefore it is either windy outside or cloudy
outside.
p p q
Proofs 17
Simplification
Prove: It is sunny and it is cold. Therefore it is sunny.
p q p
Proofs 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 qp
Proofs 19
Hypothetical syllogismProve:
If Dr. Fairbanks is speaking in chapel today, I will not skip If I don’t 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
Proofs 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
Proofs 21
Hypothetical syllogism
Prove: If you love me you will keep my commandments.
(Jn 14:15) If you keep my commandments, you will abide in my
love. (Jn 15:10) Therefore, if you love me then you will abide in my love.
This argument valid by the law of hypothetical syllogism.
Proofs 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.
Proofs 23
Proof Example
Consider:
q p q rp sr
Proofs 24
Proof Example
Consider:
p (q r) qp r
Show no way to make all premises true but conclusion false
Proofs 25
Proof by contradiction
Consider:
r sp s r q p q
Use proof by contradiction
Proofs 26
Proof Example
If the law is sufficient, then Christ died in vain
The law is sufficientTherefore Christ died in vain.
Proofs 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
Proofs 28
Sorites
Romans 10:13-15 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.
Proofs 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
Proofs 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!
Proofs 31
Example
Babies are illogicalNobody is despised who can manage a
crocodileIllogical persons are despisedTherefore, babies cannot manage crocodiles
Proofs 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.
Proofs 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.
Proofs 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.
Proofs 35
Indirect Proof of P Q
Prove the contrapositive, e.g. Q P is true, using one of the other proof methods.
Proofs 36
Proof by contradiction
Assume the negation of the proposition is true, then derive a contradiction.
Thus to prove of P Q, assume P Q is true, then derive a contradiction.
Proofs 37
Proof by cases of P Q
To prove P Q, find a set of propositions P1, P2, ..., Pn, n2, 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.
Proofs 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 didn’t die, thus thus the
premise must be false. Thus P Q must be true.
Proofs 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.
Proofs 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 evenProof:
1. Remember: x is even a:x = 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·(a+b)5. By 1, 2·(a+b) is even - QED.
Proofs 41
Direct Proof
Consider: Every multiple of 6 is also a multiple of 3.
Rewrite: x zy:(6·x = y 3·z = y)Proof:
1. Assume 6x = y (hypothesis)
2. 6x = y can be rewritten as 3 · 2x = y
3. Let z = 2x, then 3·z = y holds. QED.
Proofs 42
Indirect Proofs
Prove the contrapositive, e.g. Prove that:
Q P is true
Proofs 43
Indirect Proofs Prove: If x2 is even, then x is even.Rewrite: x : (EVEN(x2) EVEN(x))
Proofs 44
Indirect ProofsProve: 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 a:x = 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
Proofs 45
Proof by contradiction
To prove of P Q, assume P Q, derive a contradiction.
Recall that: P Q P Q
Then: P QP Q) P Q (Demorgan’s)
Thus to prove P Q we assume P Q and show a contradiction.
Proofs 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) PRIME(y) x < y)
Proofs 47
Proof by contradictionThere 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 r+1 by any prime, it will have remainder 13. r+1 is prime, since any number not divisible by any prime which is
less must be prime.4. but r+1 > p , which contradicts that p is the greatest prime number.
QED.
Proofs 48
Proof by cases
To prove P Q, find a set of propositions P1, P2, ..., Pn, n2, 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.
Thus:P(P1P2...Pn) and (P1Q)(P2Q)...(PnQ)(PQ)
Proofs 49
Proof by cases
Consider: For every nonzero integer x ,x2 > 0.
Let:P = "x is a nonzero integer”Q = x2 > 0
We want to prove P Q
Proofs 50
Proof by cases If: P = "x is a nonzero integer”
Q = x2 > 0Prove P QP can be broken up into two cases:
P1 = x > 0
P2 = x < 0
Note that P (P1 P2).
Proofs 51
Proof by cases
For every nonzero integer x ,x2 > 0. Prove each case -
Prove P1 Q:
If x > 0, then x2 > 0, since the product of two positive numbers is always positive.
Prove P2 Q:If x < 0, then x2 > 0, since the product of two negative numbers is always positive. QED.