Proofs — basic strategies for proving universal statements (CSCI 2824, Spring 2015)

It is often enlightening to learn from mistakes. On this page, we study examples of wrong proofs (and hopefully learn to not repeat them in homework/exams!).

Flawed Mathematical Arguments

We will now see examples of flawed arguments that you need to watch out for when doing mathematics. Examples include

• Proving by example: Just present a few examples and note that an universal statement holds based on these.
• Assuming some fact in the proof that does not follow from the premise.
• Proving by intuition: Appeal to your intuition usually by drawing a diagram.
• Statements that should never be present in a good proof:
  • it is clear to me (proof by vision!! )
  • it just feels right (the touchy-feely proof!! )
  • intuitively,… (proof by intuition!! )
  • my professor told us that,… (proof by authority!! )
  • it is obvious. (unless it truly truly truly is rather obvious as 1+1 = 2 !!).
• Proving the wrong statement.
  • Proof of the converse (instead of proving , we prove ).
  • Proof of something unrelated.
• Not proving the statement for all the cases.
  • Missing out on some corner cases.
    • Proof only applies to even but not to odd.

Example# 1

Claim: For , if is even, then is prime. I.e, .

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 even.

Let us attempt one more proof of this:

Assume is prime. We will prove that must be even.

• Since is prime, and , must necessarily be odd.
• Since is odd, it must be the case that is even.
• If is even, we have proved previously that is also even.
• Therefore, is even. QED??

Are there any flaws in either of these proofs? Do they convince you of the truth of our “claim”?

Answer

The 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 #2

Claim If two numbers and are odd, then is even.

Exercise: Write this down in logical notation.

Let us look at a proof:

Proof Here are our reasoning steps:

1. Since is odd, it can be written as for some .
2. Since is odd, it can be written as too for some .
3. Therefore .
4. But is an even number.
5. Therefore, is even. QED.

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?

Here are our reasoning steps:

1. Since is odd, it can be written as for some .
2. Since is odd, it can be written as too.
3. Therefore .
4. .
5. Therefore, . QED.

Can you correct the demonstrations above? What went wrong.

Answer

The 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.

Proof Here are our reasoning steps:

1. Since is odd, it can be written as for some .
2. Since is odd, it can be written as too for some .
3. Therefore .
4. But is an even number.
5. Therefore, is even. QED.

Example #3

Claim If is natural number then is a composite number.

Proof: Let be a natural number.

1. We can write as a product of .
2. Therefore is a composite number. QED??

Answer

The 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:

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 .