86 lines
2.3 KiB
Plaintext
86 lines
2.3 KiB
Plaintext
|
import Mathlib.Data.Finset.Card
|
|||
|
|
import Mathlib.Data.Set.Finite
|
|||
|
|
|
|||
|
|
/-! # Common.Set.Finite
|
|||
|
|
|
|||
|
|
Additional theorems around finite sets.
|
|||
|
|
-/
|
|||
|
|
|
|||
|
|
namespace Set
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
A set `A` is equinumerous to a set `B` (written `A ≈ B`) if and only if there is
|
|||
|
|
a one-to-one function from `A` onto `B`.
|
|||
|
|
-/
|
|||
|
|
def Equinumerous (A : Set α) (B : Set β) : Prop := ∃ F, Set.BijOn F A B
|
|||
|
|
|
|||
|
|
infix:50 " ≈ " => Equinumerous
|
|||
|
|
|
|||
|
|
theorem equinumerous_def (A : Set α) (B : Set β)
|
|||
|
|
: A ≈ B ↔ ∃ F, Set.BijOn F A B := Iff.rfl
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
For any set `A`, `A ≈ A`.
|
|||
|
|
-/
|
|||
|
|
theorem equinumerous_refl (A : Set α)
|
|||
|
|
: A ≈ A := by
|
|||
|
|
refine ⟨fun x => x, ?_⟩
|
|||
|
|
unfold Set.BijOn Set.MapsTo Set.InjOn Set.SurjOn
|
|||
|
|
simp only [imp_self, implies_true, Set.image_id', true_and]
|
|||
|
|
exact Eq.subset rfl
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
For any sets `A` and `B`, if `A ≈ B`. then `B ≈ A`.
|
|||
|
|
-/
|
|||
|
|
theorem equinumerous_symm [Nonempty α] {A : Set α} {B : Set β}
|
|||
|
|
(h : A ≈ B) : B ≈ A := by
|
|||
|
|
have ⟨F, hF⟩ := h
|
|||
|
|
refine ⟨Function.invFunOn F A, ?_⟩
|
|||
|
|
exact (Set.bijOn_comm $ Set.BijOn.invOn_invFunOn hF).mpr hF
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
For any sets `A`, `B`, and `C`, if `A ≈ B` and `B ≈ C`, then `A ≈ C`.
|
|||
|
|
-/
|
|||
|
|
theorem equinumerous_trans {A : Set α} {B : Set β} {C : Set γ}
|
|||
|
|
(h₁ : A ≈ B) (h₂ : B ≈ C)
|
|||
|
|
: ∃ H, Set.BijOn H A C := by
|
|||
|
|
have ⟨F, hF⟩ := h₁
|
|||
|
|
have ⟨G, hG⟩ := h₂
|
|||
|
|
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
If two sets are equal, they are trivially equinumerous.
|
|||
|
|
-/
|
|||
|
|
theorem eq_imp_equinumerous {A B : Set α} (h : A = B)
|
|||
|
|
: A ≈ B := by
|
|||
|
|
have := equinumerous_refl A
|
|||
|
|
conv at this => right; rw [h]
|
|||
|
|
exact this
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
A set is finite if and only if it is equinumerous to a natural number.
|
|||
|
|
-/
|
|||
|
|
axiom finite_iff_equinumerous_nat {α : Type _} {S : Set α}
|
|||
|
|
: Set.Finite S ↔ ∃ n : ℕ, S ≈ Set.Iio n
|
|||
|
|
|
|||
|
|
/--
|
|||
|
|
A set `A` is not equinumerous to a set `B` (written `A ≉ B`) if and only if
|
|||
|
|
there is no one-to-one function from `A` onto `B`.
|
|||
|
|
-/
|
|||
|
|
def NotEquinumerous (A : Set α) (B : Set β) : Prop := ¬ Equinumerous A B
|
|||
|
|
|
|||
|
|
infix:50 " ≉ " => NotEquinumerous
|
|||
|
|
|
|||
|
|
@[simp]
|
|||
|
|
theorem not_equinumerous_def : A ≉ B ↔ ∀ F, ¬ Set.BijOn F A B := by
|
|||
|
|
apply Iff.intro
|
|||
|
|
· intro h
|
|||
|
|
unfold NotEquinumerous Equinumerous at h
|
|||
|
|
simp only [not_exists] at h
|
|||
|
|
exact h
|
|||
|
|
· intro h
|
|||
|
|
unfold NotEquinumerous Equinumerous
|
|||
|
|
simp only [not_exists]
|
|||
|
|
exact h
|
|||
|
|
|
|||
|
|
end Set
|