In the language of mathematical induction, I need to show that for all n N the s

Question

In the language of mathematical induction, I need to show that for all n N the statement Pn = {n spaceships have the same engine}, is true.

a) Basis of induction: n = 1: In this case the statement P1 is trivially true as we have only one spaceship.

b) Assume now that Pn is true for some n N. I have to show that Pn+1 is also true and I will be done. Since for Pn+1 we have n + 1 spaceships, consider the rst n of them. They have the same engine by our inductive hypothesis. Now also consider the last n of them. They also have the same engine again by our inductive hypothesis. But in this way if we consider the union of these two sets of the rst n and last n of the n+1 spaceships, we have all n+1 spaceships for which we have shown that they have the same engine, so we are done.

Question: Is the above proof correct? If not, where is the mistake?

Explanation / Answer

Pn is true that is n spaceship have n engine .

Now for n+1 ,will have to show that n+1 spaceship have n+1 engine .

Since Pn is true n spaceship have n engine and since P1 is also true so one spaceship have one engine

In Pn+1 , n+1= (n)+1 for first n spaceship there is n engine for last one spaceship there is one engine

So for n+1 spaceship n+1 engine.

Pn+1 is true

Pn is true for all n.

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