Page116 64 What is wrong with the following proof by mathema

Page116, #64

What is wrong with the following \"proof\" by mathematical induction? We will prove that all computers are built by the same manufacturer. In particular, we will prove that in any collection of n computers where n is a positive integer, all of the computers are built by the same manufacturer. We first, prove P(1), a trivial process, because in any collection consisting of one computer, there is only one manufacturer. Now we assume P(k); that is, in any collection of k computers, all the computers were built by the same manufacturer. To prove P(k + 1), we consider any collection of k + 1 computers. Pull one of these k + 1 computers (call it HAL) out of the collection. By our assumption the remaining k computers all have the same manufacturer. Let HAL change places with one of these k computers. In the new group of k computers all have the same manufacturer. Thus, HAL\'s manufacturer is the same one that produced all the other computers, and all k + 1 computers have the same manufacturer.

Solution

if you number first k computers as c1,...ck

For P(k+1), when we pull HAL from k+1 computers,

There are two cases: HAL is ck+1, when all the other k computers will be initial collection from P(k)

But in the other case, HAL can ci(i not equals k+1), then we can\'t say that k computers {c1,c2...ck+1}-{ci} are manufactured by the same person.

So our assumption is wrong that we are assuming P(k) based on the quantity(number of computers) where as it has to be on quality(attribute : Manufacturer name).