Cor1 - Jean-Romain HEU

P:∀x∈R,∀y∈R,[x+y < 0 =⇒(x < 0y < 0) ].
P
R C P
R:∀x∈R,∀y∈R,[ (x < 0y < 0) =⇒x+y < 0 ].
C:∀x∈R,∀y∈R,[ (x≥0y≥0) =⇒x+y≥0 ].
C x y x ≥0y≥0
x+y≥0 [ (x≥0y≥0) =⇒x+y≥0 ]
x y C
P
R
¬R:∃x∈R,∃y∈R,[ (x < 0y < 0) x+y≥0 ].
x=−1y= 2
R
f
P:∃T∈R∗
+,∀x∈R, f(x+T) = f(x).
f
f f T
f x Rf(x) = x2
P
P¬P:∀T∈R∗
+,∃x∈R, f(x+T)6=f(x).
T∈R∗
+x= 0 f(x)=0 f(x+T) = T2T > 0
f(x+T)>0f(x+T)6=f(x)¬P
P

f x Rf(x) = cos(πx
2)−5 sin(πx 3)
P f
T= 12 x f(x+T) = f(x)x∈R
f(x+T) = cos(π(x+ 12)
2)−5 sin(π(x+ 12) 3) = cos(πx 2+ 6π)−5 sin(πx 3+ 4π).
cos sin 2π
f(x+T) = cos(πx
2)−5 sin(πx 3) = f(x).
P f
T= 12
f
cos sin cos(π(x+T)
2) cos(πx 2+
2kπ)k T
sin(πx
3)T
T= 12
a b
u v au +bv = 1
∀n∈N,∀a∈N,∀b∈N,[ (a|n b|n P GCD(a, b) = 1) =⇒ab|n].
a, b n a b n a b
ab n
k l n =ak n =bl a b
u v
au +bv = 1
anu +bnv =n n bl ak n =ablu +bakv =
ab(lu +kv)
lu +kv ab n
ak =bl =n b ak a b
b k ab ak =n
n= 12 a= 6 b= 4
a b n ab = 24 n

n n2−2n−1>0
n∈Nn≥3n2−2n−1 = (n−1)2−2n≥3n−1≥2
(n−1)2≥4 (n−1)2−2≥2n2−2n−1>0
n2n≥n2
n
24= 16 42= 16 24≥42
n= 4
n2n≥n2(n+ 1)2=
n2+ 2n+ 1 = n≥3 2n+ 1 < n2
(n+ 1)2<2n2(n+ 1)2<2×2n= 2n+1
2n+1 ≥(n+ 1)2n+ 1
2n≥n2
n≤3
n= 0,1n= 3
n= 0,1,2
n= 3
n= 0,1
n= 3 n= 2 n= 3
n= 3
X4+ 2X3−X2+X−1
aZ
a4+ 2a3−a2+a−1=0 a(a3+ 2a2−a+ 1) = 1
−1a= 1
a=−1−1 14+ 2 ×13−12+ 1 −1 = 2 6= 0
(−1)4+ 2 ×(−1)3−(−1)2+ (−1) −1 = −46= 0