From 34b60db77561ce94ced0b1e882f22da456a9c68d Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Wed, 6 Nov 2013 08:31:04 +0100
Subject: [PATCH 1/1] modif partiel S1 13

---
 partiels/131105S1/controle.log |  12 ++++++------
 partiels/131105S1/controle.pdf | Bin 88472 -> 88576 bytes
 partiels/131105S1/controle.tex |  13 +++++++++----
 3 files changed, 15 insertions(+), 10 deletions(-)

diff --git a/partiels/131105S1/controle.log b/partiels/131105S1/controle.log
index 8c27c7d..b4f0a8e 100644
--- a/partiels/131105S1/controle.log
+++ b/partiels/131105S1/controle.log
@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.3)  4 NOV 2013 20:22
+This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/Debian) (format=pdflatex 2013.11.3)  6 NOV 2013 08:26
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -1102,11 +1102,11 @@ Non-PDF special ignored!
 Non-PDF special ignored!{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
 (./controle.aux) ) 
 Here is how much of TeX's memory you used:
- 10058 strings out of 495002
- 148793 string characters out of 6180262
+ 10060 strings out of 495002
+ 148819 string characters out of 6180262
  302980 words of memory out of 5000000
- 13008 multiletter control sequences out of 15000+600000
- 15942 words of font info for 47 fonts, out of 8000000 for 9000
+ 13010 multiletter control sequences out of 15000+600000
+ 16956 words of font info for 49 fonts, out of 8000000 for 9000
  14 hyphenation exceptions out of 8191
  44i,7n,64p,599b,199s stack positions out of 5000i,500n,10000p,200000b,80000s
 {/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc}</
@@ -1119,7 +1119,7 @@ xmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texlive/texmf-dist
 public/amsfonts/symbols/msbm10.pfb></usr/share/texlive/texmf-dist/fonts/type1/u
 rw/times/utmb8a.pfb></usr/share/texlive/texmf-dist/fonts/type1/urw/times/utmr8a
 .pfb>
-Output written on controle.pdf (1 page, 88472 bytes).
+Output written on controle.pdf (1 page, 88576 bytes).
 PDF statistics:
  45 PDF objects out of 1000 (max. 8388607)
  32 compressed objects within 1 object stream
diff --git a/partiels/131105S1/controle.pdf b/partiels/131105S1/controle.pdf
index 8f6d5735fe9959a6e0a399e92a6ef0b4581ac356..e0a9fe5e3e087088d9722841dec009670f0a9ebb 100644
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diff --git a/partiels/131105S1/controle.tex b/partiels/131105S1/controle.tex
index 057bddb..1f6ac44 100644
--- a/partiels/131105S1/controle.tex
+++ b/partiels/131105S1/controle.tex
@@ -63,7 +63,7 @@ suivants:
 \item s'il a moins de 50 ans et mesure plus de 20 m de hauteur;
 \item s'il est tordu. 
 \end{itemize}
-Pour un arbre quelconque, on définit les variables booléennes suivantes par:
+Pour un arbre quelconque, on définit les variables booléennes suivantes:
 \begin{itemize}
 \item $a=1$ si l'arbre est un résineux;
 \item $b=1$ si l'arbre a moins de 50 ans;
@@ -114,7 +114,9 @@ $P(A_0) \cup P(B_0)$.
 
 
 Sur $\N^*$ on définit la relation
-$a \mathcal{R} b$ si   $a^b \leq b ^a$.
+$a \mathcal{R} b$   si et seulement si
+$a^b \leq b ^a$.
+
 \begin{enumerate}
 \item A-t-on $2 \mathcal{R} 3$? $2 \mathcal{R} 7$? $2 \mathcal{R} 4$? $4 \mathcal{R} 2$? Justifier à chaque fois.
 
@@ -122,9 +124,12 @@ $a \mathcal{R} b$ si   $a^b \leq b ^a$.
 
 \item La fonction logarithme népérien $\ln: \R^{+*} \rightarrow \R$ étant 
   croissante, montrer que sur $\N^*$
-  $a \mathcal{R} b$ si   $\dfrac{\ln(a)}{a} \leq \dfrac{\ln(b)}{b}$.
+  $$
+  a \mathcal{R} b \textrm{ si et seulement si }
+  \dfrac{\ln(a)}{a} \leq \dfrac{\ln(b)}{b}.
+  $$
 \item La relation est-elle transitive? Pour justifier votre réponse, on 
-  pourra utiliser  la question précédente.
+  pourra utiliser la question précédente.
 \item La relation est-elle antisymétrique? Le justifier.
 \end{enumerate}
 
-- 
2.39.5