Il faut vs il suffit, CN vs CS (d`après concours ESIEE 2004) Négation

P
P
P
P
P
P
P
P
P
1≤x < 3
(a, b) = (0,0) a b a b
ab
ab = 0 a b a b
(a, b)6= (0,0)
x y z P
z=x+y x < 0y= 1
P z =x+y x ≥0y6= 1
P z =x+y x ≥0y6= 1
P x < 0y= 1 z=x+y
P x < 0y= 1 z=x+y
P x < 0y= 1 z=x+y
A a N n
∃x∈Z; 2x=−7
∃x∈R; 2x=−7
∀x∈R;x2≥ −1
∀x∈R,∃y∈R;y≥x
∃y∈R,∀x∈R;y≥x

∃x∈R,∀y∈R;y2≥x
∀A > 0,∃N > 0,∀n > N; 2n+ 3 > A
∃N > 0,∀A > 0,∀n > N; 2n+ 3 > A
∀A > 0,∃N > 0,∀n > N;n2> A
∀A > 0,∃N > 0,∀n > N;1
n+ 5 > A
∀a > 0,∃N > 0,∀n > N;¯
¯
¯
¯
1
n+ 5¯
¯
¯
¯
< a
∀²∈R∗
+,∃α∈R∗ +,∀x∈R;¡|x|< α ⇒ |x2|< ²¢
n n n + 1
n n n + 1
n n −1n+ 1
x x2<4x < 2x > −2
x x2<4x < 2x > −2
x x2>4x > 2x > −2
x x2>4x > 2x < −2
a b c
ab > 1a+b≤1
a+1 b
ab > 1a+b≤1
a+1 b
µab ≥1a+b≤1
a+1 b¶⇔ab = 1
abc > 1a+b+c≤1
a+1 b+1 c