3. Find the mistakes in the following \"proof\" by mathematical induction.
\"Theorem:\" For all integers $n \ge 2$, $3^n - 2$ is even.
\"Proof (by mathematical induction): Suppose the theorem is true for an integer $k$, where $k \ge 2$.
That is, suppose that $3^k - 2$ is even. We must show that $3^{k+1} - 2$ is even. But
$3^{k+1} - 2 = 3^k \cdot 3 - 2 = 3^k(1 + 2) - 2 = (3^k - 2) + 3^k \cdot 2$.
Now $3^k - 2$ is even by the inductive hypothesis and $3^k \cdot 2$ is even by inspection. Hence the sum of the
two even quantities is even. It follows that $3^{k+1} - 2$ is even, which is what we needed to show.\"3. Is {1, 5, 4}, {7, 2}, {1, 3, 4}, {6, 8} a partition of {1, 2, 3, 4, 5, 6, 7, 8}?
