Corrigé - Jean

P Q
P:∀a∈N,∀b∈N, ab >10 =⇒a > 3b > 3.
Q:∀a∈N,∀b∈N, ab >10 =⇒a > 3b > 3.
P
C R P
C:∀a∈N,∀b∈N, a 63b63 =⇒ab < 10.
R:∀a∈N,∀b∈N, a > 3b > 3 =⇒ab >10.
N Q
N:∃a∈N,∃b∈N, ab >10 [a63b63].
P C R Q
C P
a∈Nb∈Na63b63a b
ab 63×3ab < 10 C
P
R
a= 4 b= 2 a > 3b > 3ab < 10
Q

a= 6 b= 2 ab >10 b63
P a ∈N
b∈Nab >10 a > 3b > 3
a > 3a63
a > 3a b
a63ab >10 ab 63b ab >10 3b>10
b>10
3>9 3= 3 a>3b > 3
a > 3b > 3P
C
fR
f
∀x∈R,∀y∈R,[f(x)61] [x<y =⇒f(x)< f(y)].
f1:x7→ 1
1+x2, f2:x7→ xp|x|, f3:x7→ −e−x.
f1
x=−1y= 2 f1(x) = 1
2f1(y) = 1 5x < y
f1(x)> f1(y)
f2
f2(4) = 8 >1
f3x∈R
e−x>0
f3(x) = −e−x<0f3(x)61
f3x y Rx < y
−x > −y e−x>
e−y−e−x<−e−yf3(x)< f3(y)
f3
a b c
∀d∈N, d|a d|b d|c=⇒d= 1.

pgcd(a, b)c
d pgcd(a, b)c d pgcd(a, b)
pgcd(a, b)a b d
a b d a b c
d= 1
pgcd(a, b)c
a b pgcd(a, b)c
U V W aU +bV +cW = 1
u v au +bv =
pgcd(a, b)pgcd(a, b)c
r s pgcd(a, b)r+cs = 1 (au +bv)r+cs = 1
U=ur V =vr W =s aU +bV +cW = 1
a= 10 b= 12 c= 15
U V W
a b
b−a= 2 = pgcd(a, b)
c c −7×2 = 1
c−7(b−a)=7a−7b+c= 1
n
nlog2(n) = ln(n)
ln(2)
n
n= 1 nlog2(1) = 0
n= 1
n>2k∈[[1, n −1]]
nlog2(n)
n n n >2 log2(n)>1
n n 6= 1 n
a b n =ab a < n b < n
alog2(a)blog2(b)
n a b
log2(a) + log2(b) log2(a) + log2(b) = log2(ab) = log2(n)
nlog2(n)
n