Q: Prove the proposition P(1) , where P(a) is the proposition “If n is a positive integer, then n² ≥ n.” What kind of proof did you use?

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- Jan 28th 2009, 07:17 PMGrillakisDiscrete Math - Last Problem. Need Help!
Q: Prove the proposition P(1) , where P(a) is the proposition “If n is a positive integer, then n² ≥ n.” What kind of proof did you use?

- Jan 29th 2009, 02:14 AMclic-clacQuote:

P(**a**) is the proposition “If n is a positive integer, then n² ≥ n.”

If it's and not , is a positive integer and , therefore is true. - Jan 30th 2009, 06:05 PMGrillakis
- Jan 31st 2009, 01:58 AMclic-clac
Mhh ok no problem :)

First I wanted to know :

Quote:

where P(a) is the proposition “If n is a positive integer, then n² ≥ n.”

- Jan 31st 2009, 08:32 AMGrillakis
- Jan 31st 2009, 09:36 AMclic-clac
Ok! I'll write instead of

We have

Thus

Consider the formula if is false or if is true, then is true.

is true, therefore the implication is true: we've proved

***********

Another way to prove (and even more) is to say that the proposition is a tautology (i.e. is always true). Indeed:

If isn't a positive integer, then is false and the implication is true: is true.

If is a positive integer, because the multiplication on is compatible with the order. So the second part of the implication is true, and again is true.

In conclusion, whatever is (a number, a matrix, a topological space...), is true. In particular, is true. - Jan 31st 2009, 10:03 AMGrillakis