matemáticas

P❘❖❇▲❊▼❆❙ ❉❊▲ ❈❖◆❚■◆❯❖
▼❛r❝❤ ✸✵✱ ✷✵✵✾

❯◆■❱❊❘❙■❉❆❉ ❉■❙❚❘■❚❆▲ ❋❘❆◆❈■❙❈❖ ❏❖❙➱ ❉❊ ❈❆▲❉❆❙

P❘❖❨❊❈❚❖ ❈❯❘❘■❈❯▲❆❘ ▲❊❇❊▼

❘❊❆▲■❩❆❉❖ P❖❘✿ ❉■❆◆❆ ●Ó▼❊❩✱ ▲❆❉❨ ❆▼❆❨❆✱ ❙■◆❉❨ ❈❖❘❚❊❙✱

▲❆❯❘❆ ❉❮❆❩✳

✶

❊♥ ❡st❡ ❞♦❝✉♠❡♥t♦✱ ♣r❡s❡♥t❛♠♦s ❡❧ ✐♥s✉♠♦ ❞❡ ❧❛ ♣r✐♠❡r❛ ♣❛rt❡ ❞❡❧ ❝✉rs♦ ❞❡
♣r♦❜❧❡♠❛s ❞❡❧ ❝♦♥tí♥✉♦✱ ❞✐r✐❣✐❞♦ ♣♦r ❡❧ ❞♦❝❡♥t❡ ❆❧❞♦ ■✈á♥ P❛rr❛✱ ❞❡❧ q✉❡ s♦♠♦s
✐♥t❡❣r❛♥t❡s✳ ➱st❡ ❝✉rs♦✱ ❤❛❝❡ ♣❛rt❡ ❞❡ ❡❧ ❡❥❡❞❡ Pr♦❜❧❡♠❛s ❡♥ ♥✉❡str❛ ❢♦r♠❛❝✐ó♥
❝♦♠♦ ❧✐❝❡♥❝✐❛❞❛s ❡♥ ▼❛t❡♠át✐❝❛s ❝♦♥ é♥❢❛s✐s ❡♥ ❡❞✉❝❛❝✐ó♥ ❜ás✐❝❛✳

❉❊▼❖❙❚❘❆❈■Ó◆✿
❙■

N≈N×N

❉❡✜♥❛♠♦s f

❡♥t♦♥❝❡s

N≈N×N

f :N→N×N

❡s ❜✐②❡❝t✐✈❛

: N×N → N

Esta f unci´
on se ha def inido mediante el estudio de la diagonal propuesta por Cantor.

❈♦♠♦ ❡s f ❜✐②❡❝t✐✈❛✱ ♣♦r ❝♦♠♦❞✐❞❛❞ ✉s❛r❡♠♦s ❧❛ ❢✉♥❝✐ó♥ ❛sí✿
f :N×N→N
(a, b) →

(x +y)(x +y+1)
2

✰y

❆❤♦r❛ ❝♦♠♦f ❡s ❜✐②❡❝t✐✈❛✱ ❡s ✐♥②❡❝t✐✈❛ ② s♦❜r❡②❡❝t✐✈❛✳

i ✮Demostraci´o n :f es inyectiva
❙✉♣♦♥❣❛♠♦s (x, y) =(r, s)✱ (x, y)∧ (r, s)∈ N × N
♣❆

♣r♦❜❛r q✉❡

f (x , y)=f (r , s)

❆♥❛❧✐❝❡♠♦s✱ ❧❛s ♣♦s✐❜❧❡s ❝♦♥❞✐❝✐♦♥❡s ♣❛r❛ q✉❡ (x, y)s❡❛ ❞✐❢❡r❡♥t❡ ❞❡
(r, s)②✱ ♣r♦❜❡♠♦s ✐♥②❡❝t✐✈✐❞❛❞ ♣❛r❛ ❝❛❞❛ ✉♥❛ ❞❡ ❡❧❧❛s✳
a)x = r ∧ y = s
b)x = r ∧ y = s
c) x = r ∧ y = s

a)

Pr✉❡❜❛ ♣❛r❛❆ ♣r♦❜❛r

x =r ∧ y =s

f (x , y) = f (r , s)

✷

❙✐ ♣ ❡s ❝✐❡rt♦✱ ❡♥t♦♥❝❡s ❛❧ r❡st❛r ✉♥❛ ❞❡ ❧❛ ♦tr❛ t✐❡♥❡ q✉❡ ❤❛❜❡r ✉♥
r❡s✐❞✉♦✱ ✲t❡♥✐❡♥❞♦ ❡♥ ❝✉❡♥t❛ q✉❡ [x , y, r , s ∈ N] ✲
❙✉♣♦♥❣❛♠♦s q✉❡ ❡❧ r❡s✐❞✉♦ ❞❡ ❧❛ r❡st❛ ❡s ✵
=⇒ f (x , y) = f (r , s)❧✉❡❣♦ f (x , y) − f (r, s) = 0

P❘❖❇❊▼❖❙✿
x +y(x +y+1 )
2

(r+s)(r+s+1)
2

+y−

x 2 +y 2 +2xy+x +y
2

+y−

+s =0

r 2 +s2 +2rs+r+s2

x 2 +y 2 −(r 2 +s 2 )+x +y−(r +s)+2xy−2rs
2

+s =0

+y −s =0

P❡r♦ t❡♥❡♠♦s q✉❡ x = r ② y = s ✱ ❧✉❡❣♦ x + y = r + s ❡♥t♦♥❝❡s ❛♥❛❧✐❝❡♠♦s
♣❛rt❡s ❞❡ ❧❛ s✉♠❛ ❛♥t❡r✐♦r✿
x 2 +y 2 −(r 2 +s 2 )
2

♠
⇒

x +y−(r +s)
2

=0

=0

♣♦rq✉❡ x + y = r + s ∧ r, s, y, x ∈ N

♣♦r ♠

♣❛r❛ y;y − s = 0 ♣♦rq✉❡ y = s
❆sí q✉❡✱

⇒y > s ∧ y − s = 0

x 2 +y 2 −(r 2 +s 2 )+x +y−(r+s)+2xy−2rs
2

P♦r ❡st❛ r❛③ó♥
❝♦♥tr❛❞✐❝❝✐ó♥

f (x , y) − f (r , s) = 0

❧✉❡❣♦ f (x , y) = f (r , s)

b)Pr✉❡❜❛
♣❆

♣❛r❛

♣r♦❜❛r q✉❡

❚❡♥❡♠♦s q✉❡
P❡r♦ ❝♦♠♦

x =r∧ y =s

f (x , y) = f (r , s)

x = r ⇒x + y = r + y

y =s ⇒x+y =r+s

▲✉❡❣♦ x + y + 1

=r +s +1

✸

+y −s =0

−→←−❤❡♠♦s

❧❧❡❣❛❞♦ ❛ ✉♥❛

②✱ ♦(x + y) (x + y + 1 ) = (r + s) (r + s + 1 ) ❉❡ ❡st❡ ♣❛s♦ ❤❛r❡♠♦s ✉s♦ ♣❛r❛♦tr♦s ♣✉♥t♦s ❞❡ ést❛ ♠✐s♠❛ ❞❡♠♦str❛❝✐ó♥✱ ❛sí q✉❡✱ ❧♦ ❞❡♠♦str❛r❡♠♦s ❛❧ ✜✲
♥❛❧✐③❛r❧❛✳
⇒

(x +y)(x +y+1 )
2

(r +s)(r +s+1 )
2

=

▲✉❡❣♦ (x+y)(x+y+1)
+y =
2
P❡r♦ y = s ⇒

(r+s)(r+s+1)
2

(x +y)(x +y+1 )
2

+y =

+y

(r +s)(r +s+1 )
2

+s

⇒ f (x, y) = f (r, s)

c)Pr✉❡❜❛

♣❛r❛

x =r ∧ y=s

P❛r❛ ❡st❛ ♣❛rt❡ ❞❡ ❧❛ ❞❡♠♦str❛❝✐ó♥ ✉s❛r❡♠♦s ❧❛ ♠✐s♠❛ ♠❡t♦❞♦❧♦❣í❛
q✉❡✉s❛♠♦s ♣❛r❛ a)✱ ♣❡r♦ ❡st❛ ✈❡③ ❞❡ ❛❝✉❡r❞♦ ❛❧ ❛♥á❧✐s✐s q✉❡ s❡ r❡❛❧✐③♦
s♦❜r❡ ❧❛ ❞✐❛❣♦♥❛❧ ❞❡ ❈❛♥t♦r✱ q✉❡ ❤❛ s✐❞♦ ✉s❛❞❛ ♣❛r❛ ❡❧ ♣❧❛♥t❡❛♠✐❡♥t♦
❞❡ ❧❛ ❢✉♥❝✐ó♥ q✉❡ s❡ ❤❛ ❡st❛❜❧❡❝✐❞♦❀ ② q✉❡ ♣✉❡❞❡ ❡♥❝♦♥tr❛rs❡ ❡♥
✭♠✉ñ♦③✱ ✷✵✵✷✮✳ P❛r❛ ❞❡♠♦str❛r ❡st❡ ❤❡❝❤♦ ❝♦♥s✐❞❡r❛♠♦s ❞♦s ❝❛s♦s✿
c1 (x , y) , (r , s)❡stá♥
c2 (x , y) , (r , s)♥♦

P❆❘❆

❡♥ ❧❛ ♠✐s♠❛ ❞✐❛❣♦♥❛❧✳

❡stá♥ ❡♥ ❧❛ ♠✐s♠❛ ❞✐❛❣♦♥❛❧

c1 P❘❖❇❊▼❖❙✿

x= r , y = s ∧ (x, y) , (r, s)❡stá♥

❡♥ ❧❛ ♠✐s♠❛ ❞✐❛❣♦♥❛❧

s❛❜❡♠♦s q✉❡ x + y = r + s ♣♦r ❡❧ ❡st✉❞✐♦ ❤❡❝❤♦ ❛ ❧❛ ❞✐❛❣♦♥❛❧ ♣r♦♣✉❡st❛
♣♦r ❈❛♥t♦r ❡♥t♦♥❝❡s x + y + 1 = r + s + 1 ∧ (x+y)(x+y+1)
= (r+s)(r+s+1)
2
2
❆❧ s✉♠❛r
(r+s)(r+s+1)
2

y ∧ s ❛ ❝❛❞❛ ❧❛❞♦
+ s ♣♦rq✉❡ y = s

r❡s♣❡❝t✐✈❛♠❡♥t❡✱

⇒ f (x, y) = f (r, s)

P❆❘❆

c2 P❘❖❇❊▼❖❙✿

x =r ∧ y =s ⇒ x+y =r+s

❆ ♣r♦❜❛r q✉❡✿

f (x, y) = f (r , s)

✹

⇒

(x+y)(x+y+1)
2

+y =

❚❡♥❡♠♦s q✉❡ f (x , y) = f (r , s)✱ ▲✉❡❣♦ f (x , y) >
②❛ q✉❡ {x, y, r, s ∈ N}
Pr♦❜❡♠♦s s✉♣♦♥✐❡♥❞♦ q✉❡ ❡❧ r❡s✐❞✉♦ ❡s
⇒
∗

(x+y)(x+y+1)
2

+y−

②

2

+y−s=0

2

❉✐❣❛♠♦s q✉❡
+y

2

)−(r

2

+s

2

)

2

0

+s=0

(x2 +y2 )−(r2 +s2 )+(x+y)−(r+s)+y−s+2xy−2rs

♣❡r♦ s❛❜❡♠♦s q✉❡
(x

(r+s)(r+s+1)
2...