Evaluate the Proof: All dogs are the same breed. Proof. We will prove this propo

Question

Evaluate the Proof:

All dogs are the same breed.

Proof. We will prove this proposition using mathematical induction. For each natural number n, we let P(n) be:

Any set of n dogs consists entirely of dogs of the same breed.

We will prove that for each natural number n, P(n) is true, which will prove that all dogs are the same breed. A set with only one dog consists entirely of dogs of the same breed and, hence, P(1) is true. So we let k be a natural number and assume that P(k) is true, that is, that every set of k dogs consists of dogs of the same breed. Now consider a set D of (k+ dogs, where )

D = {d1, d2, ..., dk, dk+1}.

Explanation / Answer

In this proof when trying to prove P(k+1) is true it is assumed that any k dogs from set D are of same breed. This violates the Principle of Mathematical Induction. In POMI we consider that P(k) is true then using it we need to prove that P(k+1) is true.

But here from set D d1 is eliminated which we cannot eliminate as it is part of P(k) statement. So the proof is incorrect.

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