Proving √2 is Irrational: Demonstrating with Math

[charlie_sheep]

Hello, I’m Charlie and I’ve been trying to understand how I prove things correctly. So here is an example of the way I demonstrate:

Proof that square root of 2 isn’t rational

Consider the following hypotheses
1.(√2 is rational) → (√2 = m/n | m,n ∈ ℤ, gdc(m,n) = 1) → (m² = 2n², n = (m²)/2 | m,n ∈ ℤ, gdc(m,n) = 1)
2.(∀x,y ∈ ℤ, (x² = 2y²) → (x = 2z | z ∈ ℤ))
3.(m = 2k, n = 2p | m,n,k,p ∈ ℤ, gdc(m,n) = 1) → (gcd(m,n) = 1 ∧ gdc(m,n) ≠ 1)
4.~(gcd(m,n) = 1 ∧ gdc(m,n) ≠ 1)
By 1 and 2 there is
5.(√2 is rational) → (m = 2k, n² = 2k² | m,n,k ∈ ℤ, gdc(m,n) = 1)
By 5 and 2 there is
6.(√2 is rational) → (m = 2k, n = 2p | m,n,k,p ∈ ℤ, gdc(m,n) = 1)
By 6 and 3 there is
7.(√2 is rational) → (gcd(m,n) = 1 ∧ gdc(m,n) ≠ 1)
By 7 and 4 there is
8. ~(√2 is rational)
∎

Is it correct? If not, why? Any other observations?

Note: English is not my first language, and I’m not really good in it. So, I apologize for any mistakes.

 

[Valkarie]

The proof is mostly correct. There are some minor errors and typos, but the overall structure of the proof is correct. Here are some items to consider:1. You can simplify your proof by eliminating redundant steps. For example, you can eliminate step 5. 2. You should also make sure that your notation is consistent. For example, you use “gcd” in one step (Step 3) and “gdc” in another (Step 4). This can lead to confusion when reading the proof. 3. You should also make sure that your quantifiers are clearly defined. For example, in step 2, you state that “(∀x,y ∈ ℤ, (x² = 2y²) → (x = 2z | z ∈ ℤ))”. However, it is not clear what the universal quantifier “∀x,y ∈ ℤ” applies to. It could apply to the entire statement, or just to the first part “(x² = 2y²)”. It is best to be as explicit as possible when writing proofs.