Math-info fiche de TD numéro 3

◦
(1 = 2) ⇒(2 = 3)
(a)∃x∈R∀y∈Rx+y > 0 ; (b)∀x∈R∃y∈Rx+y > 0 ;
(c)∀x∈R∀y∈Rx+y > 0 ; (d)∃x∈R∀y∈Ry2> x.
a b c d
fR R
x∈Rf(x)61
f
f
x∈R+f(x)60
⇔,⇐,⇒.
x∈Rx2= 4 . . . . . . x = 2
z∈Cz=z . . . . . . z ∈R
x∈Rx=π . . . . . . e2ix = 1
P, Q, R, S
P⇒Q
P Q
P Q R
P Q R
P Q ⇒(R⇒S)
x y z z < x
z < x + 1

∀ε > 0∃α > 0/|x−7/5|< α ⇒ |5x−7|< ε
¡P∧QV(¬P)∨Q¢
P
(¬Q)∧P⇒ ¬S
S⇒(¬P)∨Q
P⇒R∨S
S∧Q⇒ ¬P
R∧ ¬(S∨Q)⇒T
R⇒(¬P)∨(¬Q)
T
(∀x)(∃n)/(x6n)
(∃M)/(∀n)(|un|6M)
(∀x)(∀y)(xy =yx)
(∀x)(∃y)/(yxy−1=x)
(∀ε > 0)(∃N∈N)/(∀n>N)(|un|< ε)
(∀x∈R)(∀ε > 0)(∃α > 0)/(∀f∈ F)(∀y∈R)(|x−y|< α V|f(x)−f(y)|< ε)
P(x)x∈X P ={x∈X/P (x)}
P Q ¬P , P ∧Q, P ∨Q, P VQ, P ⇔Q
π O deg C
A⇒(B⇒A)
(A⇒B)∧A∧ ¬B
(A∨ ¬B)∧(A∨ ¬C)∧(B∨C)
(A∨ ¬B)∧(A∨ ¬C)∧(B∨C)∧(¬B∨ ¬A)∧(¬C∨ ¬A)
¬ ∨ ⇒
> ⊥

◦
A⇒B B (A)A(1 = 2)
(A)
B(A)A⇒B A
B
∀x∈R∃y∈Rx+y60
x∈Ry∈Rx+y60
y=−(x+ 1) x+y=x−x−1 = −160
x y =−x+ 1
x+y= 1 >0∃x∈R∀y∈Rx+y60
∀x∈R∀y∈Rx+y > 0x=−1y= 0
∃x∈R∃y∈Rx+y60
x=−1∀x∈R∃y∈Ry26x
x∈Rf(x)61
x∈Rx∈Rf(x)61
f(x)>1x∈R, f(x)>1
f
(x1, x2)x16x2f(x1)6f(x2)
x1x2x16x2f(x1)6f(x2)
(x1, x2)
x16x2f(x1)> f(x2)
x1∈Rx2∈Rx16x2f(x1)> f(x2)
f
f f
x∈R, f(x)<0x1∈R
x2∈Rx16x2f(x1)> f(x2)x∈R, f(x)<0
x∈R+f(x)60
x∈R+
f(x)>0x∈R+
f(x)>0
⇐
⇔
⇒

P Q
Q P ⇒Q
P Q R
P Q R
P Q R S
∀x∈Z∃y∈Z∀z∈Z(z
∃x∈Z∀y∈Z∃z∈Z(z
∃ε > 0∀α > 0 (|x−7/5|< α |5x−7|>ε).