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 $\displaystyle n$ and not $\displaystyle a$ , $\displaystyle 1$ is a positive integer and $\displaystyle 1^2=1\geq 1$ , therefore $\displaystyle P(1)$ 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 $\displaystyle \mathbb{N}^*$ instead of $\displaystyle \mathbb{N}-\{0\}.$

We have $\displaystyle P(n):\ n\in\mathbb{N}^*\Rightarrow n^2\geq n$

Thus $\displaystyle P(1):\ 1\in \mathbb{N}^*\Rightarrow 1^2\geq 1$

Consider the formula $\displaystyle F:\ A\Rightarrow B,$ if $\displaystyle A$ is false or if $\displaystyle B$ is true, then $\displaystyle F$ is true.

$\displaystyle 1^2\geq 1$ is true, therefore the implication $\displaystyle 1\in \mathbb{N}^*\Rightarrow 1^2\geq 1$ is true: we've proved $\displaystyle P(1).$

***********

Another way to prove $\displaystyle P(1)$ (and even more) is to say that the proposition $\displaystyle P(x)$ is a tautology (i.e. is always true). Indeed:

$\displaystyle \circ$ If $\displaystyle x$ isn't a positive integer, then $\displaystyle x\in\mathbb{N}^*$ is false and the implication $\displaystyle x\in\mathbb{N}^*\Rightarrow x^2\geq x$ is true: $\displaystyle P(x)$ is true.

$\displaystyle \circ$ If $\displaystyle x$ is a positive integer, $\displaystyle (x\geq x\ \text{and}\ x\geq 1)\Rightarrow x\times x\geq x\times 1\Rightarrow x^2\geq x$ because the multiplication on $\displaystyle \mathbb{N}$ is compatible with the order. So the second part of the implication is true, and again $\displaystyle P(x)$ is true.

In conclusion, whatever is $\displaystyle x$ (a number, a matrix, a topological space...), $\displaystyle P(x)$ is true. In particular, $\displaystyle P(1)$ is true. - Jan 31st 2009, 10:03 AMGrillakis