Mistakes in Proofs

There are many common errors made in constructing mathematical proofs. We will briefly describe some of these here.

Examples

1. What is wrong with this famous supposed “proof” that 1 = 2?

“Proof:” We use these steps, where a and b are two equal positive integers.

Solution:

Every step is valid except for one , step 5 where we divided both sides by (a – b) . The error is that (a – b) equals zero; division of both sides of an equation by the same quantity is valid as long as this quantity is not zero.

2. Suppose you want to prove the assertion:

Let a, b, ∈ Z where a = 1 mod 3 and b = 2 mod 3. Then (a + b) = 0 mod 3.

Incorrect Proof:

Since a = 1 mod 3 there is an integer k in Z such that a = 3k + 1. Since b = 2 mod 3, we can write b = 3k + 2. Thus a + b = (3k + 1) + (3k + 2) = 6k + 3 = 3(2k + 1), so (a + b) = 0 mod 3.

Error in the Proof:

The attempted proof assumes that b = a + 1 since b − a = (3k + 2) − (3k + 1) = 1. So the proof is only valid for that limited set of choices for a and b.

Correct Proof:

Since a = 1 mod 3 there is an integer k in Z such that a = 3k+1. Since b = 2 mod 3, there is an integer n in Z such that b = 3n + 2. Therefore a + b = (3k + 1) + (3n + 2) = 3k + 3n + 3 = 3(k + n + 1), so (a + b) = 0 mod 3

A proof is a valid argument that establishes the truth of a mathematical statement. A proof can use the hypothesis of the theorem, if any, axioms assumed to be true, and previously proven theorems. Using these ingredients and rules of inference, the final step of the proof establishes the truth of the statement being proved. But here we will mainly focus on more informal proof.