Chapter 12.5, Problem 38E

To determine

To Find:Check whether the given process is correct.

Explanation of Solution

Given, $P\left(n\right)$ denote the statement, ‘In any group of $n$ cats, if any one cat is black, then they are all black.’

For $n=1$ ,

If one cat is black, then all the cats are black (i.e. one).

So therefore, $P\left(n\right)$ is True for $n=1$ .

Consider $n=2$ ,

Then one of it be black, named ‘Tadpole’. Now removing some other cat (name it- Sparky). So we have left with one cat which is Tadpole, using above $P\left(n\right)$ is True for $n=1$ .So the statement is True.

Now add the Sparky into the group and remove Tadpole, so again we have left with one cat (Sparky) which is not black. Therefore, the Sparky need not be black. So, by induction hypothesis with group of 2 cats, if one cat is black (Tadpole) then all other need not be black (Sparky).

So, the given $P\left(n\right)$ statement is not True for all $n$ .

The given Induction Hypothesis is True for all $n\ge 3$ .