Writing Proofs | Proof by Contradiction Example 1

We prove a statement using the method of proof by contradiction.

[xoppa]

0:47 Nice video. One small critique. To prove a conditional statement via a 'proof by contradiction', we should assume the hypothesis is true and the conclusion is false. Symbolically a proof by contradiction has the following scheme or form.
Claim: p → q.
Proof. Assume ~ ( p → q ), or assume p & ~q since ~ ( p → q ) = p & ~q .
Then if some steps later we derive a contradiction, by the 'proof by contradiction' proof method we can conclude ~ ~ ( p → q ).
But the latter is equivalent to p → q, by the double negation rule.

[Shawty-fisn]

i proved it using quadratic nonresidues since it implies a^2 congruent to 3 mod 9 which is impossible. does this work.