So, i'm having a little problem on interpretating a result that i'm obtaining regarding a proof by contrapositive. Suppose we want to prove, by contrapositive, the following statements:
Let $S$ = $\{2, 3, 4\}$, and let $n$ $\in$ $S$. If $\frac{n^2(n-1)^2}{4}$ is even, then $\frac{n^2(n+1)^2}{4}$ is even.
$\underline{Proof}$:(by contrapositive)
Assuming $n$ is odd, $n$ can be written as $n = 2k + 1$, where $k \in \mathbb{Z}$. Since in $S$ we have not that many elements in $S$, we can verify for every element $n \in S$ that what we are trying to prove is true. So let's do the cases:
- $n = 2$: $\frac{2^2(2+1)^2}{4} = \frac{4.3^2}{4} = 9$, which is odd;
- $n = 3$: $\frac{3^2(3+1)^2}{4} = \frac{9.4^2}{4} = 36$, which is even;
- $n = 4$: $\frac{4^2(4+1)^2}{4} = \frac{4^2.5^2}{4} = 100$, which is even;
From our goal, now we want to take $n = 2 \in S$, and verify if $\frac{n^2(n-1)^2}{4}$ is odd.
Then, $\frac{2^2.(2-1)^2}{4} = \frac{4.1^2}{4} = 1$, which is odd.
Finally, the implication holds for $n = 2$. $\square$
My question now is the following: when we try to do this proof by construction, we see that the direct implication does not hold for $n = 2$. With that in mind, i've been trying to understand two things:
What can we interpret from this, knowing that the proof by construction, that is, $P \Rightarrow Q$ is logically equivalent to $\sim Q \Rightarrow \sim P$?
I was hoping that when we would try $\{3, 4\}$ would satisfy the direct implication, but only $n = 4 \in S$ holds for this $(\Rightarrow)$ direction. Why is this the case, given that $n = 2$ satisfies the contrapositive implication?