A proof is a structured argument from an initial hypotheses to a conclusion such that each step in the proof follows the laws of logic and is one of the ways to find the truth. It is especially useful in abstract cases because there is no empirical evidence and we can only argue based on logic. In logic proofs, the hypotheses are called premises and must be already proven in order to be deemed useable and the conclusion is generally, the statement you are trying to prove(not true in some cases such as proof by contradiction).
Here is an example of a proof we did in class:
For all n is in N, if n is odd, n^2 is odd
Assume n is in N
Assume n is odd
Then there exists j is in N such that n = 2j + 1
Then n^2 = (2j + 1)^2
= 4j^2 + 4j + 1
= 2(2j^2 + 2j) + 1
Then there exists k = 2j^2 + 2, n^2 = 2k + 1
Then n^2 is odd
Then n is odd implies n^2 is odd
Then for all n is in N, if n is odd, n^2 is odd
We can also use this result to prove n^2 is even implies n is even since that is the contrapositive of our result. This format resembles python code due to the emphasis on the indentation and I like it more than the format for mathematical proofs because it feels like point form but still conveys the entire meaning of the proof.
Here is an example of a proof we did in class:
For all n is in N, if n is odd, n^2 is odd
Assume n is in N
Assume n is odd
Then there exists j is in N such that n = 2j + 1
Then n^2 = (2j + 1)^2
= 4j^2 + 4j + 1
= 2(2j^2 + 2j) + 1
Then there exists k = 2j^2 + 2, n^2 = 2k + 1
Then n^2 is odd
Then n is odd implies n^2 is odd
Then for all n is in N, if n is odd, n^2 is odd
We can also use this result to prove n^2 is even implies n is even since that is the contrapositive of our result. This format resembles python code due to the emphasis on the indentation and I like it more than the format for mathematical proofs because it feels like point form but still conveys the entire meaning of the proof.
