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#1 2009-10-08 02:25:55
- Bladito
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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 
#2 2009-10-08 02:47:42
- bobbym
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Re: Equal positive integers
Hi Bladito;
Bladito wrote:
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 (2009-10-08 02:54:13)
In mathematics, you don't understand things. You just get used to them.
Probability is the most important concept in modern science, especially as nobody has the slightest notion what it means.
90% of mathematicians do not understand 90% of currently published mathematics.
#3 2009-10-08 07:34:16
- Avon
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Re: Equal positive integers
bobbym wrote:
Bladito wrote:
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.