Math Is Fun Forum

Bladito

Member

Registered: 2009-10-07

Posts: 1

Equal positive integers

Theorem: All positive integers are equal.

Proof: Sufficient to show that for any two positive integers, A and B, A = B.

Further, it is sufficient to show that for all N > 0, if A and B (positive integers) satisfy (MAX(A, B) = N) then A = B.

Proceed by induction.

If N = 1, then A and B, being positive integers, must both be 1. So A = B.

Assume that the theorem is true for some value k. Take A and B with MAX(A, B) = k+1. Then MAX((A-1), (B-1)) = k. And hence (A-1) = (B-1). Consequently, A = B.

Anyone got a solution? Cuz I have no idea

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Equal positive integers

Hi Bladito;

Assume that the theorem is true for some value k.

One problem is right there, you can't assume that. You can't find any k where that is true to start the inductive process. For any value of k > 1, I will always have the counterexample Max( A = k, B = k-1) = k for k >1, clearly A > B. This holds for all k > 1. In other words the theorem doesn't hold for 2,3,4,5 ... Max(A,B) = k does not imply A = B = k.

Last edited by bobbym (2009-10-07 03:54:13)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Avon

Member

Registered: 2007-06-28

Posts: 80

Re: Equal positive integers

Assume that the theorem is true for some value k.

One problem is right there, you can't assume that. You can't find any k where that is true to start the inductive process.

The statement is true when k=1 as Bladito has stated. You seem to agree.

Bladito, the problem is that A-1 and B-1 are not necessarily both positive integers.