Get fucking sneaky with proofs by contradiction.

When you can’t prove that shit directly, switch up your style and prove it by contradiction.

Proof by Contradiction

Contradiction is a sneaky fucking style of proof, and is definitely one of the most useful tools to have in your bag when shit gets real.

Instead of trying to prove something directly – starting with an assumption and just moving toward your conclusion with a series of if-then statements – you pull some mathematical aikido shit and assume the fucking OPPOSITE of what you’re trying to prove, and then show that the assumption you made results in a logical contradiction.  And?  If you arrive at a contradiction, then that means the assumption your ass started with is bullshit (assuming all your logic is legit).  If your assumption isn’t true, then the opposite must be true.  Boom goes the dynamite.  Before you know it, your proof is on the fucking ground, and it never saw it coming.

Best give an example to enlighten your ass.

Statement

Let $n$ be an integer such that $n^2$ is a multiple of 3.  Then $n$ is also a multiple of 3.

Proof

I love this shit.

Let’s get our contradiction on.  Step 1 is assuming the opposite of what we want to prove.  We want to prove that $n$ is a multiple of 3.  So we assume that $n$ is not a fucking multiple of 3.

Well shit, if $n$ isn’t a multiple of 3, then it’s either 1 or 2 more than a multiple of 3.  What now?  Every damn integer falls into 3 categories: a multiple of 3, a multiple of 3 plus 1, or a multiple of 3 plus 2.

…-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6…

Alright fine then, so what if $n=3x+1$?  That means…

$$n^2=(3x+1)^2$$

$$n^2=9x^2+6x+1$$

$$n^2=3(3x^2+2x)+1$$

Ooooooh shit, son!  See the problem?!  We just showed that $n^2$ is one more than a multiple of 3.  But our statement said that $n^2$ is a multiple of 3.  So we know $n\neq3x+1$

Alright but we ain’t done.  What if $n=3x+2$?  I know you didn’t forget about that scenario.  Let’s do the same shit.

$$n^2=(3x+2)^2$$

$$n^2=9x^2+12x+4$$

$$n^2=9x^2+12x+3+1$$

$$n^2=3(3x^2+4x+1)+1$$

Damn this feels good.  Same shit.  Same contradiction.  So we know $n\neq3x+2$.  Well shit, if $n$ isn’t one more than a multiple of 3, and it isn’t 2 more than a multiple of 3, then it has to be a multiple of 3 itself.  Hang on…

CONTRA.  FUCKING.  DICTION.

Keep up, now.  We assumed in the beginning that $n$ was not a multiple of 3, but we just fucking said that it had to be a multiple of 3.  That’s a goddamn contradiction, which means our initial assumption was false.  Therefore, $n$ is a multiple of 3.

QED, ya nerds.

You dig?  There’s definitely flavors of the direct method in there, because you do have to know your shit and reason your way to your contradiction, but this is a different approach that for many proofs is infinitely fucking easier than using the direct method.

So if you don’t know, now you know – proof by contradiction.

Good grief this is some fine mathematics.