Proofs 1

ProofsElementary Discrete Mathematics

Jim Skon

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Proofs

Why proofs? Careful examination to determine if mistake

has been made. Convince someone else about proposition.

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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!

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Proofs

Proofs must be based on some underlying set of truths which, in general, everyone believes.

Axioms or PostulatesDefinitions

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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

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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.

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Proofs - Definitions

Example definitions:

x is even a:x = 2a

x is odd a:x = 2a + 1

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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.

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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.

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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

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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.

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Argument

Argument - consists of a collection of statements, called premises of the argument, followed by a conclusion statement.

A1

A2

:

An

A}Premises

Conclusion

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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.

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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

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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.

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Addition

Prove: It is windy outside. Therefore it is either windy outside or cloudy

outside.

p p q

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Simplification

Prove: It is sunny and it is cold. Therefore it is sunny.

p q p

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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

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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

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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

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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.

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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.

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Proof Example

Consider:

q p q rp sr

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Proof Example

Consider:

p (q r) qp r

Show no way to make all premises true but conclusion false

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Proof by contradiction

Consider:

r sp s r q p q

Use proof by contradiction

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Proof Example

If the law is sufficient, then Christ died in vain

The law is sufficientTherefore Christ died in vain.

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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

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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.

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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

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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!

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Example

Babies are illogicalNobody is despised who can manage a

crocodileIllogical persons are despisedTherefore, babies cannot manage crocodiles

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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.

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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.

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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.

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Indirect Proof of P Q

Prove the contrapositive, e.g. Q P is true, using one of the other proof methods.

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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.

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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.

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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.

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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.

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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.

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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.

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Indirect Proofs

Prove the contrapositive, e.g. Prove that:

Q P is true

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Indirect Proofs Prove: If x2 is even, then x is even.Rewrite: x : (EVEN(x2) EVEN(x))

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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

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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.

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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)

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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.

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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)

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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

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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).

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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.