Enderton (set). Begin adding Chapter 6 lean theorems.

finite-set-exercises
Joshua Potter 2023-08-23 18:23:28 -06:00
parent 9e9072ce45
commit 306acd2975
2 changed files with 103 additions and 3 deletions

View File

@ -150,6 +150,9 @@
\lean*{Mathlib/Init/Function} \lean*{Mathlib/Init/Function}
{Function.Bijective} {Function.Bijective}
\lean{Mathlib/Logic/Equiv/Defs}
{Equiv}
\section{\defined{Equivalence Class}}% \section{\defined{Equivalence Class}}%
\hyperlabel{ref:equivalence-class} \hyperlabel{ref:equivalence-class}
@ -214,8 +217,8 @@
A set is \textbf{finite} if and only if it is \nameref{ref:equinumerous} to a A set is \textbf{finite} if and only if it is \nameref{ref:equinumerous} to a
\nameref{ref:natural-number}. \nameref{ref:natural-number}.
\lean{Mathlib/Data/Finset/Basic} \lean*{Mathlib/Data/Set/Finite}
{Finset} {Set.Finite}
\section{\defined{Function}}% \section{\defined{Function}}%
\hyperlabel{ref:function} \hyperlabel{ref:function}
@ -298,6 +301,9 @@
A set is \textbf{infinite} if and only if it is not a A set is \textbf{infinite} if and only if it is not a
\nameref{ref:finite-set}. \nameref{ref:finite-set}.
\lean*{Mathlib/Data/Set/Finite}
{Set.Infinite}
\section{\defined{Infinity Axiom}}% \section{\defined{Infinity Axiom}}%
\hyperlabel{ref:infinity-axiom} \hyperlabel{ref:infinity-axiom}
@ -3159,6 +3165,23 @@
\end{proof} \end{proof}
\subsection{\unverified{Bijections are Two-Sided Inverses}}%
\hyperlabel{sub:bijections-two-sided-inverses}
\begin{corollary}
A function $f$ is a one-to-one correspondence if and only if it has a left
and right inverse.
\end{corollary}
\begin{proof}
By definition, a one-to-one correspondence $f$ between sets $A$ and $B$ must
be both one-to-one and onto.
By \nameref{sub:theorem-3j}, $f$ is one-to-one if and only if it has a left
inverse.
The same theorem states that $f$ is onto $B$ if and only if it has a right
inverse.
\end{proof}
\subsection{\verified{Theorem 3K(a)}}% \subsection{\verified{Theorem 3K(a)}}%
\hyperlabel{sub:theorem-3k-a} \hyperlabel{sub:theorem-3k-a}
@ -8520,13 +8543,16 @@
\end{proof} \end{proof}
\subsection{\pending{Theorem 6B}}% \subsection{\verified{Theorem 6B}}%
\hyperlabel{sub:theorem-6b} \hyperlabel{sub:theorem-6b}
\begin{theorem}[6B] \begin{theorem}[6B]
No set is \nameref{ref:equinumerous} to its powerset. No set is \nameref{ref:equinumerous} to its powerset.
\end{theorem} \end{theorem}
\code*{Bookshelf/Enderton/Set/Chapter\_6}
{Enderton.Set.Chapter\_6.theorem\_6b}
\begin{proof} \begin{proof}
Let $A$ be an arbitrary set and $f \colon A \rightarrow \powerset{A}$. Let $A$ be an arbitrary set and $f \colon A \rightarrow \powerset{A}$.
Define $\phi = \{a \in A \mid a \not\in f(a)\}$. Define $\phi = \{a \in A \mid a \not\in f(a)\}$.

View File

@ -1,3 +1,5 @@
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Function import Mathlib.Data.Set.Function
import Mathlib.Data.Rel import Mathlib.Data.Rel
@ -36,6 +38,78 @@ theorem theorem_6a_c (A : Set α) (B : Set β) (C : Set γ)
: ∃ H, Set.BijOn H A C := by : ∃ H, Set.BijOn H A C := by
exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩ exact ⟨G ∘ F, Set.BijOn.comp hG hF⟩
/-- #### Theorem 6B
No set is equinumerous to its powerset.
-/
theorem theorem_6b (A : Set α)
: ∀ f, ¬ Set.BijOn f A (𝒫 A) := by
intro f hf
unfold Set.BijOn at hf
let φ := { a ∈ A | a ∉ f a }
suffices ∀ a ∈ A, f a ≠ φ by
have hφ := hf.right.right (show φ ∈ 𝒫 A by simp)
have ⟨a, ha⟩ := hφ
exact absurd ha.right (this a ha.left)
intro a ha hfa
by_cases h : a ∈ f a
· have h' := h
rw [hfa] at h
simp only [Set.mem_setOf_eq] at h
exact absurd h' h.right
· rw [Set.Subset.antisymm_iff] at hfa
have := hfa.right ⟨ha, h⟩
exact absurd this h
/-- #### Pigeonhole Principle
No natural number is equinumerous to a proper subset of itself.
-/
theorem pigeonhole_principle (m n : ) (hm : m < n)
: ∀ f : Fin m → Fin n, ¬ Function.Bijective f := by
sorry
/-- #### Corollary 6C
No finite set is equinumerous to a proper subset of itself.
-/
theorem corollary_6c (S S' : Finset α) (hS : S' ⊂ S)
: ∀ f : S → S', ¬ Function.Bijective f := by
sorry
/-- #### Corollary 6D (a)
Any set equinumerous to a proper subset of itself is infinite.
-/
theorem corollary_6d_a (S S' : Set α) (hS : S' ⊂ S) (hf : S' ≃ S)
: Set.Infinite S := by
sorry
/-- #### Corollary 6D (b)
The set `ω` is infinite.
-/
theorem corollary_6d_b
: Set.Infinite (@Set.univ ) := by
sorry
/-- #### Corollary 6E
Any finite set is equinumerous to a unique natural number.
-/
theorem corollary_6e (S : Set α) (f : S → Fin n) (hf : Function.Bijective f)
: S ≃ Fin m → m = n := by
sorry
/-- #### Lemma 6F
If `C` is a proper subset of a natural number `n`, then `C ≈ m` for some `m`
less than `n`.
-/
lemma lemma_6f {n : } (C S : Finset ) (hC : C ⊂ S) (hS : S ≃ Fin n)
: ∃ m : , m < n ∧ ∃ f : C → Fin m, Function.Bijective f := by
sorry
/-- #### Exercise 6.1 /-- #### Exercise 6.1
Show that the equation Show that the equation