Princípio de Indução Finita - Elon Lages Lima
November 18, 2022 | Author: Anonymous | Category: N/A
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X
1, 2, 3, 4, 5, . . .
N
{
1
⊂ ⊂ ∈
X N s( s (n) X
n
N
∈
∈ N
n
⊂
1 X s(X ) X = N
N
n
s(n) n + 1
s(n)
N
1
0, 1, 2, 3, 4, 5, 6, 7, 8 9
N
→ N
→N n ∈ N
∈ ⇒ ⇒
N
∈
n X
1, 2, 3, 4, 5, . . .
N
s : N s(n) N
s :
}
= 1, 2, 3, 4, 5, . . .
N
1 = s( s (n)
∈ N
n
n
s(n) 2 = s(1) 3 = s(2)
4 = s(3) 5 = s(4) 2 = s(1) 2
1
s
s
s
1
N
1
1
2
1 3
s :
→N
N
1
s
1
s
s
k
n
s(X )
⊂ X
X
⊂ N
X
1
P
s(n)
P
1
n
X X = N P
X
P
P
P (1) P (1)
→ N
P ( P (n) s(n) = s( s (s(n)) s : N s(n) = s( s (s(n)) P ( P (n) P ( P (s(n))
1
P ( P (n) n = s(n) 1
P
P
n
s(n)
X
N
→ Y
f f : N
N
→ Y
f f : X
1, 3, 5, . . . 1
∈
x X
∈
y = f = f ((x) Y
P
∈ N
N
f f : N
→ Y
f ( f (n)
n
n
∈ X ⇒ ⇒ s( s (n) ∈ X
N
n = s( s (n)
n
1 = s(1) s (1)
N
P ( P (s(n))
P
s(k )
s
−→ 3 −→ 5 −→ ···· · 2 −→ 4 −→ 6 −→ ··· · · s : N → N
s
s
1
s(1) = 3 s(3) = 5 s(5) = 7
s(n) = n + 2
{1, 2, 3, . . .}
N =
P
1
P
s
n
s
−→ 2 −→ 3 −→ 4 −→ 5 −→ ··· · ·
1
f (1) f (1)
f ( f (n)
f ( f (s(n))
n
n
f ((n) f
X
n
f
1
→ Y
f f : N
f f ((n)
1
n
2
(y , y , . . . , y , . . .) .)
· ·
k + (n + p) p) = (k + n) + p k (n p) p) =
·
k + n + n = = k k + + p p
·
k + n + n = = n n + + k k k n = n = n k
n = p p ⇒ n =
·
· ⇒
k n = k = k p
f
n
·
·
·
k (n + p + p)) = k n + k + k p
y
n = p = p
X = N
∈ X
· ·
(k n) p
∈ X ⇒ ⇒ s(n) ∈ X
f ( f (n)
n
k k
→ N
f f : N
→
k
k+n n k+ k +n
n
∈ N
k + s + s((n) = s s((k + n + n))
∈
∈
k + (n ( n + 1) = (k (k + n + n)) + 1
N
m
∈ N
m
p n
n
≤
m < n
m
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