Introduction to Proofs ch. 1.6, pg. 87,93 Muhammad Arief download dari .

Introduction to Proofsch. 1.6, pg. 87,93

Muhammad Ariefdownload dari http://arief.ismy.web.id

http://arief.ismy.web.id

Proof

Pembuktian tentang kebenaran suatu mathematical statement.

Formal proof: utilize rule of inferenceall steps were suppliedlong and hard to followsuitable for computer

Informal proof: steps maybe skiputilize assumptionsuitable for human


Proof

Example:

- Suppose you did commit the crime.

- Then at the time of the crime, you would have had to be at the scene of the crime.

- In fact, you were in a meeting with 10 people at that time, as they will testify.

- This contradicts the assumption that you committed the crime.

- Hence the assumption is false.


TOC

Direct Proof

Indirect Proof:

Proof by Contraposition

Proof by Contradiction

Exhaustive Proof

Proof Strategies


Proof

Some Important Terminology:

Theorem: a statement that can be shown to be true.

Proposition: a less important theorem.

Use “proof” to demonstrate that a theorem is true.

Axiom: statements we assume to be true.


Theorem

“For all positive real numbers x and y, if x > y, then x2 > y2”

Mathematics convention: exclude the universal quantifier

“If x > y, where x and y are positive real numbers, then x2 > y2”


Methods of Proving Theorems

p q

Proof:

Demonstrate that q is true if p is true

Methods:

Direct Proofs

Indirect Proofs:


Proof by Contradictionhttp://arief.ismy.web.id

Direct Proofs

p q

Direct Proof:- Assume that p is true- Use axiom, definition, rule of

inference etc.- Show that q must also be true


Example

Definition:The integer n is even if there exists an integer k

such that n = 2k, and n is odd if there exists an integer k such that n = 2k + 1.

Give a direct proof of the theorem:“IF n is an odd integer, THEN n2 is odd”

Solution:- Assume that “n is odd” is true- Use axiom, definition, rule of inference etc

- n = 2k + 1 (from definition)- n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1

- Conclusion: it is proved that n2 is an odd integer


ExampleDefinition:An integer a is a perfect square if there is an integer b

such that a = b2

Give a direct proof that:“IF m and n are both perfect squares, THEN nm is

also a perfect square”

Solution:- Assume that “m and n are both perfect square” is

true- Use axiom, definition, rule of inference etc

- There are integers s and t such that m = s2 and n = t2.- mn = s2t2, mn = (st)2

- Conclusion: it is proved that mn is a perfect square, because it is the square of st, which is an integer.



Indirect Proof: proof that do not start with the hypothesis and end with the conclusion.

Contraposition:p q is logically equivalent to ~q ~p.

Proof by Contraposition:- Assume that ~q is true- Use axiom, definition, rule of inference etc.- Show that ~p must also be true


Example

Prove that:“IF n is an integer and 3n + 2 is odd, THEN n

is odd”

Solution:- Assume that “n is even” is true- Use axiom, definition, rule of inference etc

- n = 2k (from definition)- 3n + 2 = 6k + 2 = 2 ( 3k + 1)- 3n + 2 is even, and therefore not odd

- Conclusion: it is proved by contrapositionhttp://arief.ismy.web.id

Proof by ContradictionContradiction:is a statement form that is always false regardless of the

truth values of the individual statements substituted for its statement variables.

q ~q~(p q) p ~q

Proof by Contradiction:- To prove that a statement p is true- Suppose the statement to be prove is false. Means its

negation is true.- Show that this supposition leads logically to a

contradiction, so the assumption is false.- Conclude that the statement to be proved is true.


Example

Prove that:“IF 3n + 2 is odd, then n is odd”

Proof by Contradiction:- p = 3n + 2 is odd, q = n is odd.- Assume that (p and ~q) is true.- Show that q is also true, OR- Show that ~p is also true.

- ~q = n is not odd = n is even- n = 2k- 3n + 2 = 6k + 2 = 2 (3k + 1) = 2 t is even- ~p is true- p and ~p are true contradiction


Example

Prove that:“There is no greatest integer”

Proof by Contradiction:- Assume that there is a greatest

integer.- Then N n for every integer n.- Let M = N + 1, M is an integer- M > N contradiction


ExampleProve that:“√2 is irrational”

Proof by Contradiction:- p = √2 is irrational.- Assume that ~p is true.- ~p = √2 is rational.- Definition: if q is rational, there exist integers a and b such

that q = a / b, where a and b has no common factor.- Then there are integers a and b with no common factors so

that √2 = a / b- 2 = a2/b2

- 2b2 = a2 Definition of even: n = 2k- Then a2 = is even a is also even- a = 2t 4t2 = 2b2 2t2 = b2 Definition of even: n = 2k- b2 is even too b is also even- a and b has common factor of 2 contradiction

- http://en.wikipedia.org/wiki/Irrational_numberhttp://arief.ismy.web.id

ExampleProve that:“1 + 3√2 is irrational”

Proof by Contradiction:- p = 1 + 3√2 is irrational.- Assume that ~p is true.- ~p = 1 + 3√2 is rational.- Definition: if q is rational, there exist integers a and b such

that r = a / b, where a and b has no common factor.- 1 + 3√2 = a/b

- 3√2 = a/b – 1- √2 = (a – b) / (3b)- By definition:

- A-b is integer- 3 b is also integer- Then √2 is rational

- Fact √2 is irrational


Exhaustive Proof

Some theorems can be proved by examining a relatively small number of examples.

Prove that (n+1)3 3n if n is a positive integer with n ≤ 4.

Solution:n = 1; (1+1)3 31 n = 2; (2+1)3 32 n = 3; (3+1)3 33 n = 4; (4+1)3 34


Proof Strategies

Analyze what the hypothesis (premises) and conclusion mean.

If it is a conditional statement:

Try a Direct Proof

Try a Proof by Contraposition

Try a Proof by Contradiction

If it has a relatively small number of domain:

Try an Exhaustive Proof


ExampleDefinition:The real number r is rational if there exists integers p

and q with q ≠ 0 such that r = p/q. A real number that is not rational is called irrational.

Prove that:“the sum of two rational number is rational.”

Direct Proof:- Assume that “r and s are both rational numbers” is

true- Use axiom, definition, rule of inference etc

- There are integers p and q, with q ≠ 0, such that r = p/q.- There are integers t and u, with u ≠ 0, such that s = t/u. - r + s = p/q + t/u = (pu + qt) / qu

- Conclusion: it is proved that r + s is rational.


Example

Prove that:“IF n is an integer and n2 is odd, THEN n is

odd”

Direct Proof:- Assume that “n2 is odd” is true- Use axiom, definition, rule of inference etc

- n2 = 2k + 1 (from definition)- n = ±√ (2k + 1)

- Conclusion: we can’t prove anything, try proof by contraposition


Example

Proof by Contraposition:- Assume that “n is not odd” is true- Use axiom, definition, rule of inference etc

- n is even- n = 2k (from definition)- n2 = 4 k2 = 2 (2 k2), therefore n2 is even (not odd)

- Conclusion: it is proved by contraposition that “IF n is an integer and n2 is odd, THEN n is odd”


Summary

Direct Proof


Proof by Contradiction

Exhaustive Proof

Proof Strategies


Introduction to Proofs ch. 1.6, pg. 87,93 Muhammad Arief download dari .

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