CSC165 Contradiction, existence, sequences

Week #5

Contradiction, existence, sequences

We started doing proofs this week. It is interesting to notice that we can structure our proofs in the same way we structure code. We can see it better by looking at an example.

In the code below, the structure and indexation is very important to understand how the code is executed.

for i in range(10):

    if i <= 5:

        print(pow(i, 3))

    else:

        print(pow(i, 1/2))

Similarly, the structure of a proof is of great importance to understand the process of the proof and to guarantee a proper use of variables and quantifiers.

∀ x ∈ ℕ, ∀ y ∈ ℕ, ∃ z ∈ ℕ, x + y = z

Assume x ∈ ℕ

            Assume y ∈ ℕ

                        Let z = x + y                        #x and y are any ordinary natural numbers

                        Then x + y = z

                        Then ∃ z ∈ ℕ, x + y = z                   

            Then ∀ y ∈ ℕ, ∃ z ∈ ℕ, x + y = z                 # Include ∀

Then ∀ x ∈ ℕ, ∀ y ∈ ℕ, ∃ z ∈ ℕ, x + y = z             # Include ∀

To sum up, precision and ingenuity are the key factors to create a good proof.

I would like to include a link to a proof that I especially like:

https://www.youtube.com/watch?v=mX91_3GQqLY