Coming at you today with a straight ballin-ass method of proof – proof by contrapositive. It’s like the direct method and the contradiction method got together and had a fucking love child. Let’s get to it.
First we need to lay down a little some logic to get your head right. Straight Spock status, son. The law of contraposition says that any conditional statement is logically equivalent to its contrapositive. To get the contrapositive of a logical statement, you essentially put your thing down, flip it, and reverse it – to put it technically. Observe:
Logical statement A: If I hear one more goddamn person says the word “twerking” again, I will honest to God slap that person across the face. I shit you not.
Fairly straightforward right? If X, then Y. Now, the contrapositive of this would be:
Contrapositive of logical statement A: If I haven’t slapped someone across the face lately, then I haven’t heard anyone utter that stupid-ass tired-ass word like people haven’t been doing that shit at high school dances for years.
Got that? If our statement is “If X, then Y,” then our contrapositive is “If not Y, then not X.”
Now, what does it mean for a statement to be logically equivalent to its contrapositive? What it means is they both say the exact same damn thing. For our purposes, it means that if you prove or disprove a statement, you prove or disprove its contrapositive, and vice versa. Statement A is true if and only if the contrapositive of A is true. Fucking slick right? To say this in some boojie-ass formatting,
$$(X\implies Y) \iff (\lnot Y\implies\lnot X)$$
Let’s make it happen, cap’n. Example time.
If the product of the integers $x$ and $y$ is odd, then both $x$ and $y$ are odd.
I could go with the direct method LIKE A FUCKING CHUMP, but I’ll go with the contrapositive like I know what the fuck I’m doing.
The contrapositive of the statement is “If both $x$ and $y$ are not odd, then $xy$ is not odd.“
Now that we have the contrapositive, we break out our direct method chops and go to town on this shit.
So since both $x$ and $y$ are not odd, then at least one of them is even. If I had to tell you that, get off my website and go watch some puppies or some shit. We’ll say that $x$ is even. So then it has a factor of 2. We’ll say $x=2k$. Alright well if that’s the case, then $xy=2ky$ AKA $xy$ is even.
Well shit, that’s what we’re trying to show, isn’t it, champ? Not going to waste my time going through this with $y$, because it would be the exact same shit we did for $x$. The choice of $x$ was arbitrary. And clearly if both $x$ and $y$ are even, then we’ll just have more factors of 2, and last I checked, that doesn’t make a number even.
So we proved just proved that the contrapositive of the original statement is true, so therefore, by the law of contraposition, the original statement is true.
Simple shit, right? The contrapositive method is a damn useful thing to have in your bag of tricks when you can’t get other methods to work right. Sometimes, though, it’s just a much cleaner way of doing it. It’s all about the elegance of it, motherfucker.
Recap of proof by contrapositive:
- Take the contrapositive of whatever you’re trying to prove.
- Prove the contrapositive.
- Be done.