What is wrong with the proof

Question

What is wrong with the following “proof” by Mathematical Induction? We will prove statements that all computers are built by the same manufacturer. In particular, we prove statements B(n) with n ∈ N, that “in any collection of n computers, all of the computers are built by the same manufacturer”. First check that B(1) is true, since with only one computer there is only one manufacturer. Now assume B(k); that is, in any collection of k computers, all are built by the same manufacturer. To prove B(k + 1) consider any collection of k + 1 computers. Pull out one of these computers, ‘HAL’ say, leaving just k computers, all of which must have the same manufacturer. Swap the ‘HAL’ computer with one of the others, so again there are k computers, so all must have the same manufacturer. Thus ‘HAL’ must have the same manufacturer as the others, hence all k + 1 computers have the same manufacturer; that is B(k + 1) must be true.

Answer 1

The inductive step actually involves $k-1$, as well as $k$, to prove $B(k+1)$, when you say that HAL and $k-1$ other computers have the same manufacturer, because the same $k-1$ computers are part of a set of $k$ earlier. So, you can go from $k$ to $k+1$ only when $k-1 > 0$, or equivalently, when $k > 1$. This is because your base case is for $k = 1$ (try to prove that a 'collection' of zero computers all have the same manufacturer!) So you can't go from 1 to 2, and the induction fails.

Answer 2

If you take out one, you get a set $A$ of $k$ computers, and if you take out another, you get a different set $B$ of $k$ computers.

But the assumption is that $A\cap B$ is not empty - that they have an element in common.

That is not true when $k+1=2$. Then $A$ and $B$ are singleton sets, and they are disjoint. So it is true that $A$ is a set all made by one manufacturer, and $B$ is a set all made by one manufacturer. To conclude that $A\cup B$ was all made by one manufacturer, $A$ and $B$ must have a common element.