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Ensembles et Applications

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Fri, 06/05/2009 - 21:35

sniper

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Joined: 06/01/2009

$On\,consid\grave{e}re\,la\,relation\,definie\,par:$

$\varphi:\large \left\{\mathbb{N}\times \mathbb{N}\to\mathbb{N}\\(x,y)\to \frac{(x+y)(x+y+1)}{2}+x\\right.$

$\,1/\,Montrer\,que\,\varphi \,determine\,une\,application$

$\,2/\,Soient\,(x,y)\,et\,(x',y')\,deux\,elements\,de\,\mathbb{N}^2\,tels\,que:\,(x,y)\neq(x',y').$

$\,On\,pose\,p=x+y\,et\,p'=x'+y'.$

$\,a)\,Montrer\,que:$

$p=p'\Rightarrow \varphi (x,y)-\varphi (x',y')=x-x'$

$\,b)\,Montrer\,que:$

$p\prec p'\Rightarrow \varphi (x,y)\prec \varphi(x',y')$

$\,c)\,En\,deduire\,que\,\varphi \,est\,une\,application\,injective.$

$\,3/\,On\,pose:$

$S(n)=1+2+........+n=\Large \sum_{i=1}^{n}{i}$

$\,Montrer\,que:$

$(\forall(x,y)\in \mathbb{N}^2):\,S(x+y)\leq \varphi (x,y)\leq S(x+y+1)$

$\,4/\,On\,considere\,l'ensemble:\,A(n)=\{p\in\mathbb{N}/ S(p)\leq n\}$

$\,a)\,Montrer\,que:$

$(\exists p_{0}\in \mathbb{N}<br /> )\,p_{0}=max(A(n))$

$\,b)\,Soit\,n\,de\,\mathbb{N}\,tels\,que:\,\varphi (x,y)=n$

$\,Montrer\,que:$

$(\exists x_{0}\in \{0,1,2.........,p_{0} \})\,n=\frac{p_{0}(p_{0}+1)}{2}+x_{0}$

$\,c)\,En\,deduire\,que\,\varphi\,est\,une\,application\,surjective.$

$\,5/\,Soit\,n\,de\,\mathbb{N},\,montrer\,que:$

$\varphi^{-1}(n)=(n-\frac{1}{2}E(\alpha_{n}-\frac{1}{2})E(\alpha_{n}+\frac{1}{2});\frac{1}{2}E(\alpha_{n}-\frac{1}{2})E(\alpha_{n}+\frac{S}{2})-n)\,/\,\alpha_{n}=\frac{1}{2}\sqrt{8n+1}$

Pythagore de Samos

Pythagore de Samos

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