CSCI 2824: Lecture Notes # 6In these notes, we are going to
Basic Proof StrategiesHow to prove a theorem of course depends on what you are asked to prove. We will give you some templates for how a proof must proceed. Of course this works only for simple theorems. For complex theorems, the idea is to decompose into simpler claims. Proving the simpler claims, we build upon them to prove more complex claims and so on. We will restrict ourselves to facts about numbers for now. Universal StatementsFor every number , some fact holds over : Proofs of Universal Statements
Universal statements are proved as follows:
Let us now look at an example. Example #1Theorem For every natural number , the number is an odd number. Proof
QED. Universal Statements With ImplicationWe now look at a special form of universal statements of the form: Universal Statement With Implication
. Following the general rule for universal statements, we write a proof as follows:
We can use a simple short-cut that avoids unnecessary language in such proofs.
Example #1Theorem If is an even number and then is composite. Proof
Here are our reasoning steps:
Example #2Theorem For every natural number , If then cannot be prime. Proof
Here are the steps of our reasoning.
Existential StatementsWe will mention existential statements. They are rarer but interesting. Simple existential statements assert that a number with some property exists. . To prove an existential statement, we just give an example. ExampleTheorem There exist two prime numbers whose sum is also a prime number. Proof
and serve as examples to our theorem. Diversion: Constructive ProofsExistential statements can be proved in another way without producing an example. Typically this involves a proof by contradiction (we will study these types of proofs soon). Such proofs are called non-constructive proofs. Theorem There exist two irrational numbers and such that is rational. Proof
We will show that such numbers exist without giving you a concrete example. Consider the number and . Therefore is rational. We know that is irrational. There are two cases:
As a result, there must exist two numbers such that is rational while themselves are irrational. ( Our argument just has not produced any concrete example to point to. :-) ) QED. Discuss some of the philosophical implications of constructivism vs. non-constructivism Needless to say we will leave non-constructive proofs to mathematicians and the debate to the philosophers for now. If you are interested, these ideas are usually covered in a philosophy of mathematics or a philosophy of science class. Flawed Mathematical ArgumentsWe will now see examples of flawed arguments that you need to watch out for when doing mathematics. Examples include
Example# 1Claim: For , If is even is prime. I.e, . Proof Attempt # 1
Let us test for , we have is 5. Works. It also works for since is prime and since is prime. Therefore, is prime if is prime. Let us attempt one more proof of this: Proof Attempt # 2
Assume is prime. We will prove that must be even.
Are there any flaws in either of these proofs? Do they convince you of the truth of our “claim”? AnswerThe claim is false in the first place because it fails for , wherein . The first proof attempt is a proof by example which is generally invalid for universally quantified statements. The second proof attempt actually sets out to prove the converse. Instead of proving is prime, it assumes this and tries to prove, instead, that is even. Example #2Claim If two numbers and are odd, then is even. Exercise: Write this down in logical notation. Let us look at a proof: Attempted Proof
Proof Here are our reasoning steps:
Is there anything wrong with the proof above? Now let us look at a related claim: Claim-2 If two numbers and are odd, then . Is this a true statement? Proof
Here are our reasoning steps:
Can you correct the demonstrations above? What went wrong. AnswerThe problem was in assuming that for some . By saying that , for some and for some , there is a flawed assumption that , which was never warranted. Therefore, we are able to “prove” Claim-2, which is clearly false. For example, and yields us and . Claim-1 is correct and the corrected proof is as follows: Claim-1 If two numbers and are odd, then is even. Corect Proof
Proof Here are our reasoning steps:
Example #3Claim If is natural number then is a composite number. Proof
Proof: Let be a natural number.
AnswerThe claim is actually false. Take , we have , a prime number. What went wrong in the proof? Well, we are correct in writing as but this does not immediately show that is composite. We have to convince ourselves that and . Recall: Definition: Prime and Composite Numbers
A natural number is composite if it can be written as for natural numbers where cannot be or itself. In logic, we define a predicate as follows: . Likewise, natural number is prime if for some natural numbers , then or . In logic, we define a predicate for natural numbers, as follows: An important exception involves the numbers . These are taken to be neither prime nor composite. The proof above can only be correct when and . |