❈❛♣ít✉❧♦ ✶
❚❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s
✶✳✶ ◆♦çõ❡s ❣❡r❛✐s
❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠ ❝♦♥❥✉♥t♦ é q✉❛❧q✉❡r ❝♦❧❡çã♦✱ ❞❡♥tr♦ ❞❡ ✉♠ t♦❞♦ ❞❡ ♦❜❥❡t♦s ❞❡✜♥✐❞♦s ❡ ❞✐st✐♥❣✉í✈❡✐s✱ ❝❤❛♠❛❞♦s ❡❧❡♠❡♥t♦s✱ ❞❡ ♥♦ss❛ ✐♥t✉✐çã♦ ♦✉ ♣❡♥s❛♠❡♥t♦✳
❖❜s❡r✈❛çã♦ ✶✳✶✳✶✳ ❊ss❛ ❞❡✜♥✐çã♦ ✐♥t✉✐t✐✈❛ ❢♦✐ ❞❛❞❛ ♣♦r ●❡♦r❣ ❈❛♥t♦r✱ ❡♠
1895✳
❉❡✜♥✐çã♦ ✶✳✶✳✷✳ ❯♠ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ ❛♣❡♥❛s ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡✲ ♠❡♥t♦s é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ✜♥✐t♦✳ ❯♠ ❝♦♥❥✉♥t♦ q✉❡ ♥ã♦ ❝♦♥té♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦✳
◆♦t❛çã♦ ✶✳✶✳✶✳ ❊♠ ❣❡r❛❧✱ ❞❡♥♦t❛r❡♠♦s ♦s ❝♦♥❥✉♥t♦s ♣❡❧❛s ❧❡tr❛s ❧❛t✐♥❛s ♠❛✐ús❝✉❧❛s ❡ ♦s s❡✉s ❡❧❡♠❡♥t♦s ♣❡❧❛s ❧❡tr❛s ❧❛t✐♥❛s ♠✐♥ús❝✉❧❛s✳
❙❡Aé ✉♠ ❝♦♥❥✉♥t♦ ❡aé ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦A,❡♥tã♦ ❡s❝r❡✈❡r❡♠♦s a ∈A✭❧❡✐❛✲s❡✿ aé ✉♠ ❡❧❡♠❡♥t♦ ❞❡ A ♦✉ ♣❡rt❡♥❝❡ ❛ A✮✳ ◗✉❛♥❞♦ a ♥ã♦ é ✉♠ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ A✱ ❡s❝r❡✈❡r❡♠♦s a /∈A✳
▼✉✐t❛s ✈❡③❡s ♦s ❝♦♥❥✉♥t♦s sã♦ ❞❡s✐❣♥❛❞♦s ❢❡❝❤❛♥❞♦✲s❡ ❡♥tr❡ ❝❤❛✈❡s ♦s sí♠❜♦❧♦s q✉❡ r❡♣r❡s❡♥t❛♠ s❡✉s ❡❧❡♠❡♥t♦s ✭q✉❛♥❞♦ é ♣♦ssí✈❡❧ ❢❛③❡r ✐ss♦✮✳ ❋r❡✲ q✉❡♥t❡♠❡♥t❡✱ ✉s❛♠♦s ❛ ❢♦r♠❛
A=
a|a ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡P
❛ q✉❛❧ ❞❡✈❡ s❡r ❧✐❞❛ ❝♦♠♦ ✧A é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s a t❛❧ q✉❡ ♣♦ss✉❡♠ ♦✉ ✈❡r✐✜❝❛♠ ❛ ♣r♦♣r✐❡❞❛❞❡ P✧✳
✷ ❈❆P❮❚❯▲❖ ✶✳ ❚❊❖❘■❆ ❉❖❙ ❈❖◆❏❯◆❚❖❙
❊①❡♠♣❧♦ ✶✳✶✳✶✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s
Z=
. . . ,−3,−2,−1,0,+1,+2,+3, . . . ❊①❡♠♣❧♦ ✶✳✶✳✷✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♣❛r❡s
2Z=
. . . ,−4,−2,0,+2,+4, . . . ♦✉
2Z=
a|a é ✉♠ ✐♥t❡✐r♦ ♣❛r . ❊①❡♠♣❧♦ ✶✳✶✳✸✳ ❖ ❝♦♥❥✉♥t♦
A=
(x, y)| x ❡ y sã♦ ♥ú♠❡r♦s r❡❛✐s s❛t✐s❢❛③❡♥❞♦ ❛ ❡q✉❛çã♦ x2
+y2
= 1 . ❆ ♣♦s✐çã♦ ✭♦r❞❡♠✮ ❡♠ q✉❡ ❛♣❛r❡❝❡♠ ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉t♥♦ ♥ã♦ t❡♠ ✐♠♣♦rtâ♥❝✐❛✳ ❆ss✐♠✱ ♦s ❝♦♥❥✉♥t♦s A =
0,±1,±2,±3, . . . ❡ Z sã♦ ♦s
♠❡s♠♦s✳
❖s ❡❧❡♠❡♥t♦s q✉❡ ❛♣❛r❡❝❡♠ ♥✉♠ ❝♦♥❥✉♥t♦ sã♦ ❞✐st✐♥t♦s✳ ❆ss✐♠✱ ❛ ♥♦t❛çã♦
a, a, b ♥ã♦ é ❛♣r♦♣r✐❛❞❛ ❡ ❞❡✈❡ s❡r s✉❜st✐t✉✐❞❛ ♣♦r
a, b ✳
❋✐♥❛❧♠❡♥t❡✱ s❡ a é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ A✱ ❡♥tã♦ a ❡
a sã♦ ♦❜❥❡t♦s ❞❡ ♥❛t✉r❡③❛ ❞✐❢❡r❡♥t❡s✳
✶✳✷ ■❣✉❛❧❞❛❞❡ ❡ s✉❜❝♦♥❥✉♥t♦s
❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❉♦✐s ❝♦♥❥✉♥t♦sA ❡B sã♦ ❝❤❛♠❛❞♦s ❞❡ ✐❣✉❛✐s ♦✉ ✐❞ê♥t✐❝♦s q✉❛♥❞♦ ❝♦♥t❡♠ ♦s ♠❡s♠♦s ❡❧❡♠❡♥t♦s✳ ❉❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣♦rA =B✳
◗✉❛♥❞♦ ❞♦✐s ❝♦♥❥✉♥t♦sA❡B ♥ã♦ ❢♦r❡♠ ✐❣✉❛✐s✱ ❞✐r❡♠♦s q✉❡ sã♦ ❞✐❢❡r❡♥t❡s ❡ ❞❡♥♦t❛r❡♠♦s ❡st❛ r❡❧❛çã♦ ♣♦rA6=B✳
❉❡✜♥✐çã♦ ✶✳✷✳✷✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦sA ❡B✳ ❉✐r❡♠♦s q✉❡Aé s✉❜❝♦♥❥✉♥t♦ ❞❡ B s❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ A é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ B✳ ❉❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣♦r A ⊆B✳
❙❡ A é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ B✱ ❡♥tã♦ ❝❤❛♠❛r❡♠♦s B ❞❡ s✉r♣❡r❝♦♥❥✉♥t♦ ❞❡ A ❡ ❞❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣♦r B ⊇A✳
◗✉❛♥❞♦ A ♥ã♦ é s✉❜❝♦♥❥✉♥t♦ ❞❡ B ✭✐st♦ é✱ B ♥ã♦ é s✉♣❡r❝♦♥❥✉♥♦ ❞❡ A✮✱ ❡s❝r❡✈❡r❡♠♦sA6⊆B ♦✉ B 6⊇A✳
❉❡✜♥✐çã♦ ✶✳✷✳✸✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦sA❡B✳ ❉✐③❡♠♦s q✉❡Aé s✉❜❝♦♥❥✉♥t♦ ♣ró♣r✐♦ ❞❡ B s❡ A ⊆B ❡ A6=B✳ ❉❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣♦r A(B✳
✶✳✸✳ ❉■❆●❘❆▼❆❙ ❉❊ ❱❊◆◆ ✸
✶✳✸ ❉✐❛❣r❛♠❛s ❞❡ ❱❡♥♥
✶✳✹ Pr♦♣r✐❡❞❛❞❡s ❞♦s s✉❜❝♦♥❥✉♥t♦s
Pr♦♣♦s✐çã♦ ✶✳✹✳✶✳ ❚♦❞♦ ❝♦♥❥✉♥t♦ é s✉❜❝♦♥❥✉♥t♦ ✭❡ ✉♠ s✉♣❡r❝♦♥❥✉♥t♦✮ ❞❡ s✐ ♠❡s♠♦✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✳ P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ a ∈ A t❡♠♦s q✉❡ a∈A✳ ▲♦❣♦✱ A⊆A✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✷✳ ❙❡❥❛♠ A✱ B ❡ C ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r✳ ❙❡ A ⊆ B ❡ B ⊆C✱ ❡♥tã♦ A⊆C✳
❉❡♠♦♥str❛çã♦✳ P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦a∈At❡♠♦s q✉❡a∈B✱ ♣♦✐sA⊆B✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ b ∈ B t❡♠♦s q✉❡ b ∈ C✱ ♣♦✐s B ⊆C✳ ▲♦❣♦✱
♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ a ∈A t❡♠♦s q✉❡a∈C✳ ❈♦♥❝❧✉í♠♦s q✉❡ A⊆C✳
Pr♦♣♦s✐çã♦ ✶✳✹✳✸✳ ❙❡❥❛♠ A ❡ B ❝♦♥❥✉♥t♦s q✉❛✐sq✉❡r✳ ❊♥tã♦✱ A =B s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A⊆B ❡ B ⊆A✳
❉❡♠♦♥str❛çã♦✳ ❙❡ A = B✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ✶✳✷✳✶✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r ❢❛✲ ❝✐❧♠❡♥t❡ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ A é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ B ❡ q✉❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ B é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ A✳ ❆ss✐♠✱ A⊆B ❡ B ⊆A✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s♦❜ ❛s
❤✐♣ót❡s❡s s❡ s✉♣♦r♠♦s q✉❡ A 6=B✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ✶✳✷✳✶✱ ♣♦❞❡♠♦s ❝♦♥✲ ❝❧✉✐r q✉❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ a∈ A q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ B ♦✉ q✉❡ ❡①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ b ∈B q✉❡ ♥ã♦ ♣❡rt❡♥❝❡ ❛ A✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ▲♦❣♦✱ ❞❡✈❡♠♦s t❡r A=B✳
✶✳✺ ❯♥✐ã♦ ❡ ✐♥t❡rs❡çã♦ ❞❡ ❝♦♥❥✉♥t♦s
❉❡✜♥✐çã♦ ✶✳✺✳✶✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉❡✜♥✐♠♦s ❛ ✉♥✐ã♦ ❞❡A ❡ B ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s x t❛✐s q✉❡ x ♣❡rt❡♥❝❡ ❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣❡❧♦ sí♠❜♦❧♦ A∪B✳ ❆ss✐♠✱
A∪B =
x| x∈A ♦✉ x∈B .
❉❡✜♥✐çã♦ ✶✳✺✳✷✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉❡✜♥✐♠♦s ❛ ✐♥t❡rs❡çã♦ ❞❡ A ❡ B ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s x t❛✐s q✉❡ x ♣❡rt❡♥❝❡ ❛ ❛♠❜♦s ♦s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉❡♥♦t❛r❡♠♦s t❛❧ r❡❧❛çã♦ ♣❡❧♦ sí♠❜♦❧♦ A∩B✳ ❆ss✐♠✱
A∩B =
✹ ❈❆P❮❚❯▲❖ ✶✳ ❚❊❖❘■❆ ❉❖❙ ❈❖◆❏❯◆❚❖❙
Pr♦♣♦s✐çã♦ ✶✳✺✳✶✳ ❙❡❥❛♠ três ❝♦♥❥✉♥t♦sA✱ B ❡C✳ ❆s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈❡r❞❛❞❡✐r❛s✿
✭✐✮ A⊆A∪B❀ ✭✐✐✮ A∪B =B∪A❀ ✭✐✐✐✮ A∪B
∪C =A∪ B∪C
❀ ✭✐✈✮ A∩B ⊆A❀
✭✈✮ A∩B =B∩A❀ ✭✈✐✮ A∩B
∩C =A∩ B∩C
❀ ✭✈✐✐✮ A∩ B∪C
= A∩B
∪ A∩C
✳
❉❡♠♦♥str❛çã♦✳ ✭✐✮ P❛r❛ t♦❞♦ x ∈ A, t❡♠♦s ❝❡rt❛♠❡♥t❡ x ∈ A∪B✳ ▲♦❣♦✱ A⊆A∪B.
✭✐✐✮ P❛r❛ t♦❞♦ x ∈ A∪B✱ t❡♠♦s q✉❡ x ∈ A ♦✉ x ∈ B ♦ q✉❡ ✐♠♣❧✐❝❛ ❝❡rt❛✲ ♠❡♥t❡ q✉❡x∈B ♦✉x∈A✳ ▲♦❣♦x∈B∪A. Pr♦✈❛♠♦s q✉❡A∪B ⊆B∪A. ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ B∪A ⊆ A∪B. ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s q✉❡A∪B =B∪A.
✭✐✐✐✮ P❛r❛ t♦❞♦ x ∈ A∪B
∪C✱ t❡♠♦s q✉❡ x ∈ A∪B ♦✉ x ∈ C✳ ❈♦♥✲
s✐❞❡r❡♠♦s ❞♦✐s ❝❛s♦s✿
1o ❝❛s♦✳
x∈A∪B✳ ❙❡x∈A✱ ❡♥tã♦ ❝❡rt❛♠❡♥t❡x∈A∪ B∪C
✳ ❙❡x∈B✱ ❡♥tã♦ x ∈ B∪C ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x ∈ A∪ B ∪C
✳ ❆ss✐♠✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦ t❡♠♦s x∈A∪ B∪C
✳
2o ❝❛s♦✳
x∈C✳ ❙❡x∈C✱ ❡♥tã♦x∈B∪C ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠x∈A∪ B∪C
✳ P♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡ A∪B
∪C ⊆A∪ B∪C
✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ x∈A∪ B∪C
✱ t❡♠♦s q✉❡ x ∈A ♦✉ x∈B ∪C✳ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ❝❛s♦s✿
1o ❝❛s♦✳
x∈A✳ ❙❡x∈A✱ ❡♥tã♦x∈A∪B ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠x∈ A∪B
∪C✳
2o ❝❛s♦✳
x ∈ B ∪C✳ ❙❡ x ∈ B✱ ❡♥tã♦ ❝❡rt❛♠❡♥t❡ x ∈ A∪B ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x∈ A∪B
∪C✳ ❙❡x∈C✱ ❡♥tã♦x∈ A∪B
∪C✳ ❆ss✐♠✱ ❡♠ q✉❛❧q✉❡r
❝❛s♦ t❡♠♦s x∈ A∪B
∪C✳
P♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡A∪ B∪C
⊆ A∪B
∪C✳ ❖s ❝❛s♦s ❛♥❛❧✐s❛❞♦s ♥♦s ♣❡r♠✐t❡♠ ❝♦♥❝❧✉✐r q✉❡ A∪B
∪C =A∪ B∪C
✶✳✻✳ ❖ ❈❖◆❏❯◆❚❖ ❱❆❩■❖ ✺
✭✐✈✮ P❛r❛ t♦❞♦ x ∈ A ∩ B, t❡♠♦s q✉❡ x ∈ A ❡ x ∈ B ❡ ❡♠ ♣❛rt✐❝✉❧❛r t❡♠♦s q✉❡ x∈A✳ ▲♦❣♦✱ A∩B ⊆A.
✭✈✮ P❛r❛ t♦❞♦ x ∈ A ∩B✱ t❡♠♦s q✉❡ x ∈ A ❡ x ∈ B ♦ q✉❡ ✐♠♣❧✐❝❛ ❝❡r✲ t❛♠❡♥t❡ q✉❡ x∈B ❡x∈A✳ ▲♦❣♦x∈B∩A.Pr♦✈❛♠♦s q✉❡A∩B ⊆B∩A. ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ B∩A ⊆ A∩B. ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s q✉❡ A∩B =B∩A.
✭✈✐✮ P❛r❛ t♦❞♦ x ∈ A ∩B
∩ C✱ t❡♠♦s q✉❡ x ∈ A ∩ B ❡ x ∈ C✳ ❉❡✲ ❝♦rr❡ ❞✐ss♦ q✉❡ [x∈A ❡ x∈B]❡ x∈C ❡ ♣♦rt❛♥t♦x∈A ❡ [x∈B ❡ x∈C]
♦ q✉❡ ❛❝❛rr❡t❛ ❡♠x∈A∩ B∩C
✳ ❆ss✐♠✱ A∩B
∩C ⊆A∩ B∩C
✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ A∩ B∩C
⊆ A∩B
∩C✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ A∩B
∩C =A∩ B∩C
✳
✭✈✐✐✮ P❛r❛ t♦❞♦ x ∈ A ∩ B ∪ C
t❡♠♦s q✉❡ x ∈ A ❡ x ∈ B ∪ C✳ ❉❡✲ ❝♦rr❡ ❞✐ss♦ q✉❡ ✭x ∈ A ❡ x ∈ B✮ ♦✉ ✭x ∈ A ❡ x ∈ C✮ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
x ∈ A∩B
∪ A∩C
✳ ▲♦❣♦✱ A∩ B ∪C
⊆ A∩B
∪ A∩C
✳ ❘❡❝✐✲ ♣r♦❝❛♠❡♥t❡✱ s❡ x ∈ A ∩B
∪ A∩C
✱ ❡♥tã♦ x ∈ A∩B ♦✉ x ∈ A∩C✳ ❈♦♥s✐❞❡r❡♠♦s ❞♦✐s ❝❛s♦s✿
1o ❝❛s♦✳
x∈A∩B✳ ◆❡ss❡ ❝❛s♦✱ t❡♠♦s q✉❡ x∈A ❡ x∈B ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x∈A ❡ x∈B∪C✱ ✐st♦ é✱x∈A∩ B∪C
✳
2o ❝❛s♦✳
x∈A∩C✳ ◆❡ss❡ ❝❛s♦✱ t❡♠♦s q✉❡ x∈A ❡ x∈C ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ x∈A ❡x∈B∪C✱ ✐st♦ é✱x∈A∩ B∪C
✳ ❆ss✐♠✱ ❡♠ q✉❛❧q✉❡r ❝❛s♦ t❡♠♦s x∈A∩ B∪C
✳ ▲♦❣♦✱ A∩B
∪ A∩C
⊆A∩ B∪C
✳ ❖s ❝❛s♦s ❛♥❛❧✐s❛❞♦s ♥♦s ♣❡r♠✐t❡♠ ❝♦♥❝❧✉✐r q✉❡A∩ B∪C
= A∩B
∪
A∩C
✳
✶✳✻ ❖ ❝♦♥❥✉♥t♦ ✈❛③✐♦
❉❡✜♥✐çã♦ ✶✳✻✳✶✳ ❯♠ ❝♦♥❥✉♥t♦ ❝♦♠ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦ é ❝❤❛♠❛❞♦ ❞❡ ❝♦♥✲ ❥✉♥t♦ ✈❛③✐♦✳
Pr♦♣♦s✐çã♦ ✶✳✻✳✶✳ ❚♦❞♦ ❝♦❥✉♥t♦ ✈❛③✐♦ é s✉❜❝♦♥❥✉♥t♦ ❞❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r ❡ V ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦✳ ❙✉✲ ♣♦♥❤❛♠♦s q✉❡ V 6⊆ A✱ ❡♥tã♦ V ⊆ A é ❢❛❧s♦✳ ■ss♦ ✐♠♣❧✐❝❛ q✉❡ ❡①✐st❡ ❛♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ v ∈ V t❛❧ q✉❡ v /∈ A✱ ♦ q✉❡ é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦✱
✻ ❈❆P❮❚❯▲❖ ✶✳ ❚❊❖❘■❆ ❉❖❙ ❈❖◆❏❯◆❚❖❙
Pr♦♣♦s✐çã♦ ✶✳✻✳✷✳ ❖ ❝♦❥✉♥t♦ ✈❛③✐♦ é ú♥✐❝♦✳
❉❡♠♦♥str❛çã♦✳ ❙❡❥❛♠V1 ❡V2 ❞♦✐s ❝♦♥❥✉♥t♦s ✈❛③✐♦s✳ ❊♥tã♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✻✳✶✱ t❡♠♦s q✉❡V1 ⊆V2 ❡ V2 ⊆V1✳ ▲♦❣♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ V1 =V2✳
◆♦t❛çã♦ ✶✳✻✳✶✳ ❉❡♥♦t❛♠♦s ♦ ❝♦♥❥✉♥t♦ ✈❛③✐♦ ♣❡❧♦ sí♠❜♦❧♦ ∅✳
❉❡✜♥✐çã♦ ✶✳✻✳✷✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉✐③❡♠♦s q✉❡ A ❡ B sã♦ ❝♦♥❥✉♥t♦s ❞✐s❥✉♥t♦s s❡A∩B =∅✳
✶✳✼ ❈♦♥❥✉♥t♦ ✉♥✐✈❡rs♦ ❡ ❝♦♠♣❧❡♠❡♥t❛r
❉❡✜♥✐çã♦ ✶✳✼✳✶✳ ❙❡❥❛ U ✉♠ ❝♦♥❥✉♥t♦✳ ❉✐③❡♠♦s q✉❡ U é ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐✲ ✈❡rs♦ s❡ ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s ♦s ❝♦♥❥✉♥t♦s ❝♦♥s✐❞❡r❛✲ ❞♦s sã♦ ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ U✳
❉❡✜♥✐çã♦ ✶✳✼✳✷✳ ❙❡❥❛ U ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐✈❡rs♦ ❡ A ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ U✳ ❉❡✜♥✐♠♦s ♦ ❝♦♠♣❧❡♠❡♥t❛r ❞❡ A ♦✉ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ A ❝♦♠♦ s❡♥❞♦ ♦ ❝♦♥✲ ❥✉♥t♦
A′ =
x| x∈U ❡ x /∈A .
❚❡♦r❡♠❛ ✶✳✼✳✶✳ ✭▲❡✐s ❞❡ ▼♦r❣❛♥✮ ❙❡❥❛ U ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐✈❡rs♦ ❡ A, B s✉❜❝♦♥❥✉♥t♦s ❞❡ U✳ ❊♥tã♦✱
✭✐✮ A∪B′
=A′∩B′❀ ✭✐✐✮ A∩B′
=A′∪B′✳
❉❡♠♦♥str❛çã♦✳ ✭✐✮ P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ x ∈ A∪B′
✱ t❡♠♦s q✉❡ x ∈ U ❡ x /∈ A∪B✳ ❙❡❣✉❡ ❞✐ss♦ q✉❡ ✭x ∈ U ❡ x /∈ A✮ ❡ ✭x ∈ U ❡ x /∈ B✮ ♦ q✉❡
❛❝❛rr❡t❛ ❡♠x∈A′ ❡x∈B′✳ ▲♦❣♦✱x∈A′∩B′✳ ❆ss✐♠✱ A∪B′
⊆A′∩B′✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦x∈A′∩B′✱ t❡♠♦s q✉❡x∈A′ ❡x∈B′✳ ❙❡❣✉❡ ❞✐ss♦ q✉❡ ✭x∈U ❡x /∈A✮ ❡ ✭x∈U ❡x /∈B✮ ♦ q✉❡ ❛❝❛rr❡t❛ ❡♠x∈U ❡x /∈A∪B✳ ▲♦❣♦✱x∈ A∪B′✳ ❆ss✐♠✱
A′∩B′ ⊆ A∪B′✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ t❡♠♦s A∪B′
=A′∩B′✳ ✭✐✐✮ ❊①❡r❝í❝✐♦✳
❚❡♦r❡♠❛ ✶✳✼✳✷✳ ❙❡❥❛ U ✉♠ ❝♦♥❥✉♥t♦ ✉♥✐✈❡rs♦✳ ❊♥tã♦✱ ✭✐✮ A′′
✶✳✽✳ ❈❆❘❉■◆❆▲✱ ❈❖◆❏❯◆❚❖ P❖❚✃◆❈■❆ ❊ P❘❖❉❯❚❖ ❈❆❘❚❊❙■❆◆❖ ❉❊ ❈❖◆❏❯◆❚❖❙✼
✭✐✐✮ ∅′ =U❀ ✭✐✐✐✮ U′ =∅✳
❉❡♠♦♥str❛çã♦✳ ❊①❡r❝í❝✐♦✳
❉❡✜♥✐çã♦ ✶✳✼✳✸✳ ❙❡❥❛♠ ❞♦✐s ❝♦♥❥✉♥t♦s A ❡ B✳ ❉❡✜♥✐♠♦s ♦ ❝♦♠♣❧❡♠❡♥t♦ r❡❧❛t✐✈♦ ❞❡ B ❡♠ A ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦
A−B =
x| x∈A ❡ x /∈B . ❉✐❛❣r❛♠❛ ❞❡ ❱❡♥♥
✶✳✽ ❈❛r❞✐♥❛❧✱ ❝♦♥❥✉♥t♦ ♣♦tê♥❝✐❛ ❡ ♣r♦❞✉t♦ ❝❛r✲
t❡s✐❛♥♦ ❞❡ ❝♦♥❥✉♥t♦s
❉❡✜♥✐çã♦ ✶✳✽✳✶✳ ❙❡❥❛ A ✉♠ ❝♦♥❥✉♥t♦ q✉❛❧q✉❡r✳ ❉❡✜♥✐♠♦s ♦ ❝❛r❞✐♥❛❧ ❞❡ A ❝♦♠♦ s❡♥❞♦ ❛ ♠❡❞✐❞❛ ❞♦ t❛♠❛♥❤♦ ❞❡ A✳ ❉❡♥♦t❛r❡♠♦s t❛❧ ❝♦♥❝❡✐t♦ ♣❡❧♦ sí♠❜♦❧♦
A
✳ ❆ss✐♠✱ ♣❛r❛ ❝♦♥❥✉♥t♦s ✜♥✐t♦s s✉❛s ❝❛r❞✐♥❛❧✐❞❛❞❡s ✐♥❞✐❝❛♠ ♦
♥ú♠❡r♦ ❞❡ ❡❧❡♠❡♥t♦s ❝♦♥st✐t✉✐♥t❡s ❞❡ s✉❛s ❝♦❧❡çõ❡s✳ P❛r❛ ❝♦♥❥✉♥t♦s ✐♥✜♥✐t♦s t❛❧ ♥♦çã♦ ♥ã♦ é tã♦ ✐♥t✉✐t✐✈❛✳
❊①❡♠♣❧♦ ✶✳✽✳✶✳ ❙❡ A=
a1, . . . , an ✱ ❡♥tã♦
A
=n✳
❊①❡♠♣❧♦ ✶✳✽✳✷✳ ❙❡ A =
a1, a2, a3, b3, b4 ❡ B =
a1, b1, b2, b3, b4 ✱ ❡♥tã♦
A∪B
= 7,
A
= 5,
B
= 5 ❡
A∩B
= 3. ▲♦❣♦✱ A∪B
= A + B −
A∩B .
❉❡✜♥✐çã♦ ✶✳✽✳✷✳ ❙❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ A✳ ❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ ❞❛s ♣❛rt❡s ❞❡ A ♦✉ ❝♦♥❥✉♥t♦ ♣♦tê♥❝✐❛ ❞❡ A ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s✉❜❝♦♥❥✉♥t♦s ❞❡ A✳
◆❡ss❡ ❝❛s♦✱ ❞❡♥♦t❛♠♦s t❛❧ ❝♦♥❥✉♥t♦ ♣❡❧♦s sí♠❜♦❧♦s ℘(A) ♦✉ 2A✳
❊①❡♠♣❧♦ ✶✳✽✳✸✳ ❙❡ A=
a1, a2, a3 ✱ ❡♥tã♦
2A
=
∅,
a1 ,
a2 ,
a3 ,
a1, a2 ,
a1, a3 ,
a2, a3 ,
a1, a2, a3
. ◆❡ss❡ ❝❛s♦✱ 2 A = 2 3 ✳
❉❡✜♥✐çã♦ ✶✳✽✳✸✳ ❙❡❥❛♠ n ❝♦♥❥✉♥t♦s A1, A2, . . . , An✳ ❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦
♣r♦❞✉t♦ ❝❛rt❡s✐❛♥♦ ❞❡A1, A2, . . . , An✱ ❞❡♥♦t❛❞♦ ♣❡❧♦ s✐♠❜♦❧♦A1×A2×. . .×An✱
❝♦♠♦ s❡♥❞♦
A1×A2×. . .×An =
a|a= (a1, a2, . . . , an)é q✉❛❧q✉❡r ❧✐st❛ ♦r❞❡♥❛❞❛
✽ ❈❆P❮❚❯▲❖ ✶✳ ❚❊❖❘■❆ ❉❖❙ ❈❖◆❏❯◆❚❖❙
❊①❡♠♣❧♦ ✶✳✽✳✹✳ ❙❡ A=
a1, a2, a3 ❡ B =
b1, b2 ✱ ❡♥tã♦ A×B =
(a1, b1),(a1, b2),(a2, b1),(a2, b2),(a3, b1),(a3, b2) . ◆❡ss❡ ❝❛s♦✱
A×B
= 3×2✳
❚❡♦r❡♠❛ ✶✳✽✳✶✳ ❙❡❥❛♠A ❡ B ❝♦♥❥✉♥t♦s ✜♥✐t♦s q✉❛✐sq✉❡r✳ ❊♥tã♦✿ ✭✐✮
A∪B =
A
+
B
−
A∩B
;
✭✐✐✮ 2
A = 2
A
❀
✭✐✐✐✮ A×B
=
A
·
B
.
✶✳✾ P❛r❛❞♦①♦s ❞❛ t❡♦r✐❛ ❞♦s ❝♦♥❥✉♥t♦s
✶✳✾✳✶ P❛r❛❞♦①♦ ❞❡ ❘✉ss❡❧
❊♠ 1902✱ ❇❡rtr❛♥❞ ❘✉ss❡❧ ♣r♦✈♦✉ q✉❡ ❛❞♠✐ssã♦ ❞❛ ✐❞é✐❛ ❞❡ ✧❝♦♥❥✉♥t♦ ❞❡
t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✧❧❡✈❛ ❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦ ♦✉ ♣❛r❛❞♦①♦✳ ❆ss✐♠✱
✧◆ã♦ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝♦♥❥✉♥t♦s✧
