Specification: Each of these proofs contains at least one flawed hypothesis (or

Question

Specification:

Each of these proofs contains at least one flawed hypothesis (or fallacy) whose introduction into the proof leads to a contradictory conclusion, rendering the proof invalid (or fallacious). For each proof, you must identify at least one fallacy and explain in detail why the fallacy leads to a contradiction. Some of the proofs contain multiple fallacies, but you only need to identify one per proof.

P4 The following will prove the property P(n) that every marble in a bag of n marbles is the same color. BASIS: In a bag of 1 marble, there is only 1 marble and it is the same color as itself, so P(1) is trivially true. INDUCTION: Assume P(k) is true, that every marble in any bag of k marbles is the same color. Let us check whether that property will necessarily follow for a bag of k + 1 marbles by drawing a marble from the bag, which we will call the first marble. The bag now contains only k marbles, so by inductive hypothesis the marbles left in the bag are all the same color. To find out that color, draw a second marble from the bag and look at it. Without loss of generality, let's say that the color observed for the second marble is red. Then every marble left in the bag is also red. Throw the first marble back into the bag. Now the bag once again has only k marbles in it, and so the first marble, which is now back in the bag, must be red just like all the other red marbles still in the bag. Throw the second marble back into the bag. Then the bag now contains k +1 marbles that are all red and, therefore, all the same color. qed

Explanation / Answer

Fallacy : The bag contains k marbles so by induction hypothesis they must all be of same colour.

Suppose a bag contains k red marbles and 1 blue marble , so in total it contains k+1 marbles.Now , in the first draw I get a red marble , then the bag contains k-1 red marbles and 1 blur marble , so in total it has k marbles which are not of same colour.

So the induction hypothesis is itseld flawed.

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