diff --git a/Bookshelf/Enderton/Set.tex b/Bookshelf/Enderton/Set.tex index bd5ddde..de10c27 100644 --- a/Bookshelf/Enderton/Set.tex +++ b/Bookshelf/Enderton/Set.tex @@ -3949,7 +3949,7 @@ Then $F = G$. \end{proof} -\subsection{\pending{Exercise 3.12}}% +\subsection{\verified{Exercise 3.12}}% \label{sub:exercise-3.12} Assume that $f$ and $g$ are functions and show that @@ -3958,6 +3958,9 @@ Assume that $f$ and $g$ are functions and show that \begin{proof} + \lean{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_12} + Let $f$ and $g$ be \nameref{ref:function}s. \paragraph{($\Rightarrow$)}% @@ -3982,7 +3985,7 @@ Assume that $f$ and $g$ are functions and show that \end{proof} -\subsection{\pending{Exercise 3.13}}% +\subsection{\verified{Exercise 3.13}}% \label{sub:exercise-3.13} Assume that $f$ and $g$ are functions with $f \subseteq g$ and @@ -3991,6 +3994,9 @@ Show that $f = g$. \begin{proof} + \lean{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_13} + Let $f$ and $g$ be functions such that $f \subseteq g$ and $\dom{g} \subseteq \dom{f}$. By \nameref{sub:exercise-3.12}, it follows that $\dom{f} \subseteq \dom{g}$ @@ -4001,7 +4007,7 @@ Show that $f = g$. \end{proof} -\subsection{\pending{Exercise 3.14}}% +\subsection{\verified{Exercise 3.14}}% \label{sub:exercise-3.14} Assume that $f$ and $g$ are functions. @@ -4014,17 +4020,22 @@ Assume that $f$ and $g$ are functions. \begin{proof} - Assume $f$ and $g$ are functions. + \statementpadding + + \lean*{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_14\_a} + + \lean{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_14\_b} + + Assume $f$ and $g$ are \nameref{ref:function}s. \paragraph{(a)}% Consider $f \cap g$. By definition of the intersection of sets, $f \cap g \subseteq f$. - By \nameref{sub:exercise-3.12}, $\dom{(f \cap g)} = \dom{f}$ and - $(\forall x \in \dom{(f \cap g)} (f \cap g)(x) = f(x)$. - The latter conjunct shows that, since $f$ is single-valued, $f \cap g$ must - also be single-valued. - In other words, $f \cap g$ is a function. + Since $f$ is single-valued, it trivially follows that so must $f \cap g$. + Therefore $f \cap g$ is a function. \paragraph{(b)}% @@ -4064,7 +4075,7 @@ Assume that $f$ and $g$ are functions. \end{proof} -\subsection{\pending{Exercise 3.15}}% +\subsection{\verified{Exercise 3.15}}% \label{sub:exercise-3.15} Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in @@ -4073,6 +4084,9 @@ Show that $\bigcup{\mathscr{A}}$ is a function. \begin{proof} + \lean{Bookshelf/Enderton/Set/Chapter\_3} + {Enderton.Set.Chapter\_3.exercise\_3\_15} + Let $\mathscr{A}$ be a set of \nameref{ref:function}s such that for any $f$ and $g$ in $\mathscr{A}$, either $f \subseteq g$ or $g \subseteq f$. Let $x \in \dom{\bigcup{\mathscr{A}}}$. diff --git a/Bookshelf/Enderton/Set/Chapter_3.lean b/Bookshelf/Enderton/Set/Chapter_3.lean index 6b19d44..955f811 100644 --- a/Bookshelf/Enderton/Set/Chapter_3.lean +++ b/Bookshelf/Enderton/Set/Chapter_3.lean @@ -325,7 +325,7 @@ theorem exercise_3_7 {R : Set.Relation α} simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this exact hxy_mem this -section +section Relation open Set.Relation @@ -678,11 +678,8 @@ theorem theorem_3k_c_ii {F : Set.Relation α} {A B : Set α} unfold image at nv simp only [Set.mem_setOf_eq] at nv have ⟨u₁, hu₁⟩ := nv - have ⟨x, hx⟩ := hF v (mem_pair_imp_snd_mem_ran hu.right) - simp only [and_imp] at hx - have hr₁ := hx.right u (mem_pair_imp_fst_mem_dom hu.right) hu.right - have hr₂ := hx.right u₁ (mem_pair_imp_fst_mem_dom hu₁.right) hu₁.right - rw [hr₂, ← hr₁] at hu₁ + have := single_rooted_eq_unique hF hu.right hu₁.right + rw [← this] at hu₁ exact absurd hu₁.left hu.left.right exact ⟨hv₁, hv₂⟩ @@ -714,6 +711,173 @@ theorem corollary_3l_iii {G : Set.Relation α} {A B : Set α} single_valued_self_iff_single_rooted_inv.mp hG exact (theorem_3k_c_ii hG').symm -end +/-- #### Exercise 3.12 + +Assume that `f` and `g` are functions and show that +``` +f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f) f(x) = g(x). +``` +-/ +theorem exercise_3_12 {f g : Set.Relation α} + (hf : f.isSingleValued) (_ : g.isSingleValued) + : f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f, ∃! y : α, (x, y) ∈ f ∧ (x, y) ∈ g) := by + apply Iff.intro + · intro h + apply And.intro + · show ∀ x, x ∈ dom f → x ∈ dom g + intro x hx + have ⟨y, hy⟩ := dom_exists hx + exact mem_pair_imp_fst_mem_dom (h hy) + · intro x hx + have ⟨y, hy⟩ := dom_exists hx + refine ⟨y, ⟨hy, h hy⟩, ?_⟩ + intro y₁ hy₁ + exact single_valued_eq_unique hf hy₁.left hy + · intro ⟨_, hx⟩ + show ∀ p, p ∈ f → p ∈ g + intro (x, y) hp + have ⟨y₁, hy₁⟩ := hx x (mem_pair_imp_fst_mem_dom hp) + rw [single_valued_eq_unique hf hp hy₁.left.left] + exact hy₁.left.right + +/-- #### Exercise 3.13 + +Assume that `f` and `g` are functions with `f ⊆ g` and `dom g ⊆ dom f`. Show +that `f = g`. +-/ +theorem exercise_3_13 {f g : Set.Relation α} + (hf : f.isSingleValued) (hg : g.isSingleValued) + (h : f ⊆ g) (h₁ : dom g ⊆ dom f) + : f = g := by + have h₂ := (exercise_3_12 hf hg).mp h + have hfg := Set.Subset.antisymm_iff.mpr ⟨h₁, h₂.left⟩ + ext p + have (a, b) := p + apply Iff.intro + · intro hp + have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp) + rw [single_valued_eq_unique hf hp hx.left.left] + exact hx.left.right + · intro hp + rw [← hfg] at h₂ + have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp) + rw [single_valued_eq_unique hg hp hx.left.right] + exact hx.left.left + +/-- #### Exercise 3.14 (a) + +Assume that `f` and `g` are functions. Show that `f ∩ g` is a function. +-/ +theorem exercise_3_14_a {f g : Set.Relation α} + (hf : f.isSingleValued) (_ : g.isSingleValued) + : (f ∩ g).isSingleValued := + single_valued_subset hf (Set.inter_subset_left f g) + +/-- #### Exercise 3.14 (b) + +Assume that `f` and `g` are functions. Show that `f ∪ g` is a function **iff** +`f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`. +-/ +theorem exercise_3_14_b {f g : Set.Relation α} + (hf : f.isSingleValued) (hg : g.isSingleValued) + : (f ∪ g).isSingleValued ↔ + (∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by + apply Iff.intro + · intro h x hx + have ⟨y₁, hy₁⟩ := hf x hx.left + have ⟨y₂, hy₂⟩ := hg x hx.right + have : y₁ = y₂ := by + have hf' : (x, y₁) ∈ f ∪ g := Or.inl hy₁.left.right + have hg' : (x, y₂) ∈ f ∪ g := Or.inr hy₂.left.right + exact single_valued_eq_unique h hf' hg' + rw [← this] at hy₂ + refine ⟨y₁, ⟨hy₁.left.right, hy₂.left.right⟩, ?_⟩ + intro y₃ hfy₃ + exact single_valued_eq_unique hf hfy₃.left hy₁.left.right + · intro h x hx + by_cases hfx : x ∈ dom f + · by_cases hgx : x ∈ dom g + · -- `x ∈ dom f ∧ x ∈ dom g` + have ⟨y₁, hy₁⟩ := hf x hfx + have ⟨y₂, hy₂⟩ := hg x hgx + refine ⟨y₁, ⟨?_, Or.inl hy₁.left.right⟩, ?_⟩ + · unfold ran + simp only [Set.mem_image, Set.mem_union, Prod.exists, exists_eq_right] + exact ⟨x, Or.inl hy₁.left.right⟩ + · intro y₃ hy₃ + apply Or.elim hy₃.right + · intro hxy + exact single_valued_eq_unique hf hxy hy₁.left.right + · refine fun hxy => single_valued_eq_unique hg hxy ?_ + have : y₁ = y₂ := by + have ⟨z, ⟨hz, _⟩⟩ := h x ⟨hfx, hgx⟩ + rw [ + single_valued_eq_unique hf hy₁.left.right hz.left, + single_valued_eq_unique hg hy₂.left.right hz.right + ] + rw [this] + exact hy₂.left.right + · -- `x ∈ dom f ∧ x ∉ dom g` + have ⟨y, hy⟩ := dom_exists hfx + have hxy : (x, y) ∈ f ∪ g := (Set.subset_union_left f g) hy + refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩ + intro y₁ hy₁ + apply Or.elim hy₁.right + · intro hx' + exact single_valued_eq_unique hf hx' hy + · intro hx' + exact absurd (mem_pair_imp_fst_mem_dom hx') hgx + · by_cases hgx : x ∈ dom g + · -- `x ∉ dom f ∧ x ∈ dom g` + have ⟨y, hy⟩ := dom_exists hgx + have hxy : (x, y) ∈ f ∪ g := (Set.subset_union_right f g) hy + refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩ + intro y₁ hy₁ + apply Or.elim hy₁.right + · intro hx' + exact absurd (mem_pair_imp_fst_mem_dom hx') hfx + · intro hx' + exact single_valued_eq_unique hg hx' hy + · -- `x ∉ dom f ∧ x ∉ dom g` + exfalso + unfold dom at hx + simp only [ + Set.mem_image, + Set.mem_union, + Prod.exists, + exists_and_right, + exists_eq_right + ] at hx + have ⟨_, hy⟩ := hx + apply Or.elim hy + · intro hz + exact absurd (mem_pair_imp_fst_mem_dom hz) hfx + · intro hz + exact absurd (mem_pair_imp_fst_mem_dom hz) hgx + +/-- #### Exercise 3.15 + +Let `𝓐` be a set of functions such that for any `f` and `g` in `𝓐`, either +`f ⊆ g` or `g ⊆ f`. Show that `⋃ 𝓐` is a function. +-/ +theorem exercise_3_15 {𝓐 : Set (Set.Relation α)} + (h𝓐 : ∀ F ∈ 𝓐, F.isSingleValued) + (h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G ∨ G ⊆ F) + : isSingleValued (⋃₀ 𝓐) := by + intro x hx + have ⟨y₁, hy₁⟩ := dom_exists hx + refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩ + intro y₂ hy₂ + have ⟨f, hf⟩ : ∃ f : Set.Relation α, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁ + have ⟨g, hg⟩ : ∃ g : Set.Relation α, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right + apply Or.elim (h f g hf.left hg.left) + · intro hf' + have := hf' hf.right + exact single_valued_eq_unique (h𝓐 g hg.left) hg.right this + · intro hg' + have := hg' hg.right + exact single_valued_eq_unique (h𝓐 f hf.left) this hf.right + +end Relation end Enderton.Set.Chapter_3 \ No newline at end of file diff --git a/Bookshelf/Enderton/Set/Relation.lean b/Bookshelf/Enderton/Set/Relation.lean index e874f57..c065cb7 100644 --- a/Bookshelf/Enderton/Set/Relation.lean +++ b/Bookshelf/Enderton/Set/Relation.lean @@ -180,7 +180,7 @@ A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, ther exists exactly one `x` such that `⟨x, y⟩ ∈ R`. -/ def isSingleRooted (R : Relation α) : Prop := - ∀ y ∈ R.ran, ∃! x, x ∈ R.dom ∧ (x, y) ∈ R + ∀ y ∈ ran R, ∃! x, x ∈ dom R ∧ (x, y) ∈ R /-- A single-rooted `Relation` should map the same output to the same input. @@ -204,7 +204,7 @@ exists exactly one `y` such that `⟨x, y⟩ ∈ R`. Notice, a `Relation` that is single-valued is a function. -/ def isSingleValued (R : Relation α) : Prop := - ∀ x ∈ R.dom, ∃! y, y ∈ R.ran ∧ (x, y) ∈ R + ∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R /-- A single-valued `Relation` should map the same input to the same output. @@ -263,6 +263,21 @@ theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α} conv => lhs; rw [← inv_inv_eq_self F] rw [single_valued_inv_iff_single_rooted_self] +/-- +The subset of a function must also be a function. +-/ +theorem single_valued_subset {F G : Set.Relation α} + (hG : G.isSingleValued) (h : F ⊆ G) + : F.isSingleValued := by + unfold isSingleValued + intro x hx + have ⟨y, hy⟩ := dom_exists hx + unfold ExistsUnique + simp only + refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hy, hy⟩, ?_⟩ + intro y₁ hy₁ + exact single_valued_eq_unique hG (h hy₁.right) (h hy) + /-- A `Relation` `R` is one-to-one if it is a single-rooted function. -/