Enderton. Exercises 3.19-3.23.

Bookshelf/Enderton/Set.tex

@@ -75,7 +75,7 @@ The \textbf{composition} of sets $F$ and $G$ is
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.comp}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.comp}
 
 \end{definition}
 
@@ -87,7 +87,7 @@ The \textbf{domain} of set $R$, denoted $\dom{R}$, is given by
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.dom}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.dom}
 
 \end{definition}
 
@@ -137,7 +137,7 @@ Given \nameref{ref:relation} $R$, the \textbf{field} of $R$, denoted $\fld{R}$,
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.fld}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.fld}
 
 \end{definition}
 
@@ -182,7 +182,7 @@ The \textbf{image of $A$ under $F$} is the set
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.image}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.image}
 
 \end{definition}
 
@@ -194,7 +194,7 @@ The \textbf{inverse} of a set $F$ is the set
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.inv}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.inv}
 
 \end{definition}
 
@@ -206,7 +206,7 @@ For any sets $u$ and $v$, the \textbf{ordered pair} $\left< u, v \right>$ is
 
 \begin{definition}
 
-  \lean*{Common/Set/OrderedPair}{OrderedPair}
+  \lean*{Bookshelf/Enderton/Set/OrderedPair}{OrderedPair}
 
 \end{definition}
 
@@ -305,7 +305,7 @@ The \textbf{range} of set $R$, denoted $\ran{R}$, is given by
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.ran}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.ran}
 
 \end{definition}
 
@@ -328,7 +328,7 @@ A \textbf{relation} is a set of \nameref{ref:ordered-pair}s.
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation}
 
 \end{definition}
 
@@ -340,7 +340,7 @@ The \textbf{restriction} of a set $F$ to set $A$ is the set
 
 \begin{definition}
 
-  \lean*{Common/Set/Relation}{Set.Relation.restriction}
+  \lean*{Bookshelf/Enderton/Set/Relation}{Set.Relation.restriction}
 
 \end{definition}
 
@@ -1341,7 +1341,8 @@ Let $A$ and $B$ be sets. $x \not\in A + B$ if and only if either
 
 \begin{proof}
 
-  \lean{Common/Set/Basic}{Set.not\_mem\_symm\_diff\_inter\_or\_not\_union}
+  \lean{Bookshelf/Enderton/Set/Basic}
+    {Set.not\_mem\_symm\_diff\_inter\_or\_not\_union}
 
   By definition of the \nameref{ref:symmetric-difference},
     \begin{align*}
@@ -2657,13 +2658,13 @@ If not, then under what conditions does equality hold?
 
   \statementpadding
 
-  \lean*{Common/Set/Relation}
+  \lean*{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.dom\_inv\_eq\_ran\_self}
 
-  \lean*{Common/Set/Relation}
+  \lean*{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.ran\_inv\_eq\_dom\_self}
 
-  \lean{Common/Set/Relation}
+  \lean{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.inv\_inv\_eq\_self}
 
   We prove that (i) $\dom{(F^{-1})} = \ran{F}$, (ii) $\ran{(F^{-1})} = \dom{F}$,
@@ -2715,10 +2716,10 @@ If not, then under what conditions does equality hold?
 
   \statementpadding
 
-  \lean*{Common/Set/Relation}
+  \lean*{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.single\_valued\_inv\_iff\_single\_rooted\_self}
 
-  \lean{Common/Set/Relation}
+  \lean{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.single\_valued\_self\_iff\_single\_rooted\_inv}
 
   We prove that (i) any set $F$, $F^{-1}$ is a function iff $F$ is
@@ -2778,7 +2779,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
 \begin{proof}
 
-  \lean{Common/Set/Relation}
+  \lean{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.one\_to\_one\_self\_iff\_one\_to\_one\_inv}
 
   We prove that (i) $F^{-1}$ is a function and (ii) $F^{-1}$ is single-rooted.
@@ -2868,7 +2869,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
   \statementpadding
 
-  \lean*{Common/Set/Relation}
+  \lean*{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.single\_valued\_comp\_is\_single\_valued}
 
   \lean{Bookshelf/Enderton/Set/Chapter\_3}
@@ -2924,7 +2925,7 @@ For any one-to-one function $F$, $F^{-1}$ is also one-to-one.
 
 \begin{proof}
 
-  \lean{Common/Set/Relation}
+  \lean{Bookshelf/Enderton/Set/Relation}
     {Set.Relation.comp\_inv\_eq\_inv\_comp\_inv}
 
   By definition of the \nameref{ref:composition} of $F$ and $G$,
@@ -4296,13 +4297,16 @@ Evaluate each of the following: $A(\emptyset)$, $\img{A}{\emptyset}$,
 
 \end{proof}
 
-\subsection{\pending{Exercise 3.20}}%
+\subsection{\verified{Exercise 3.20}}%
 \label{sub:exercise-3.20}
 
 Show that $F \restriction A = F \cap (A \times \ran{F})$.
 
 \begin{proof}
 
+  \lean{Bookshelf/Enderton/Set/Chapter\_3}
+    {Enderton.Set.Chapter\_3.exercise\_3\_20}
+
   Let $F$ and $A$ be arbitrary sets.
   By \nameref{sub:corollary-3c} and definition of the \nameref{ref:restriction},
     intersection, and \nameref{ref:range} of sets,
@@ -4319,34 +4323,38 @@ Show that $F \restriction A = F \cap (A \times \ran{F})$.
 
 \end{proof}
 
-\subsection{\pending{Exercise 3.21}}%
+\subsection{\verified{Exercise 3.21}}%
 \label{sub:exercise-3.21}
 
 Show that $(R \circ S) \circ T = R \circ (S \circ T)$.
 
 \begin{proof}
 
+  \lean{Bookshelf/Enderton/Set/Relation}
+    {Set.Relation.comp\_assoc}
+
   Let $R$, $S$, and $T$ be arbitrary sets.
   By definition of the \nameref{ref:composition} of sets,
     \begin{align*}
       (R \circ S) \circ T
         & = \{\left< u, v \right> \mid
-          \exists t(u(R \circ S)t \land tTv)\} \\
+          \exists t(uTt \land t(R \circ S)v)\} \\
         & = \{\left< u, v \right> \mid
-          \exists t((\exists a, uRa \land aSt) \land tTv)\} \\
+          \exists t(uTt \land (\exists a(tSa \land aRv))\} \\
         & = \{\left< u, v \right> \mid
-          \exists t, \exists a, (uRa \land aSt \land tTv)\} \\
+          \exists t, \exists a, (uTt \land tSa) \land aRv)\} \\
         & = \{\left< u, v \right> \mid
-          \exists a, \exists t, (uRa \land aSt \land tTv)\} \\
+          \exists a, \exists t, (uTt \land tSa) \land aRv)\} \\
         & = \{\left< u, v \right> \mid
-          \exists a(uRa \land (\exists t, aSt \land tTv))\} \\
+          \exists a, (\exists t(uTt \land tSa)) \land aRv)\} \\
-        & = \{\left< u, v \right> \mid \exists a(uRa \land a(S \circ T)v)\} \\
+        & = \{\left< u, v \right> \mid
+          \exists a, u(S \circ T)a \land aRv)\} \\
         & = R \circ (S \circ T).
     \end{align*}
 
 \end{proof}
 
-\subsection{\pending{Exercise 3.22}}%
+\subsection{\verified{Exercise 3.22}}%
 \label{sub:exercise-3.22}
 
 Show that the following are correct for any sets.
@@ -4360,6 +4368,17 @@ Show that the following are correct for any sets.
 
 \begin{proof}
 
+  \statementpadding
+
+  \lean*{Bookshelf/Enderton/Set/Relation}
+    {Enderton.Set.Chapter\_3.exercise\_3\_22\_a}
+
+  \lean*{Bookshelf/Enderton/Set/Relation}
+    {Enderton.Set.Chapter\_3.exercise\_3\_22\_b}
+
+  \lean{Bookshelf/Enderton/Set/Relation}
+    {Enderton.Set.Chapter\_3.exercise\_3\_22\_c}
+
   Let $A$, $B$, $F$, $G$, and $Q$ be arbitrary sets.
 
   \paragraph{(a)}%
@@ -4408,7 +4427,7 @@ Show that the following are correct for any sets.
 
 \end{proof}
 
-\subsection{\pending{Exercise 3.23}}%
+\subsection{\verified{Exercise 3.23}}%
 \label{sub:exercise-3.23}
 
 Let $I_A$ be the identity function on the set $A$.
@@ -4418,6 +4437,14 @@ Show that for any sets $B$ and $C$,
 
 \begin{proof}
 
+  \statementpadding
+
+  \lean*{Bookshelf/Enderton/Set/Relation}
+    {Enderton.Set.Chapter\_3.exercise\_3\_23\_i}
+
+  \lean{Bookshelf/Enderton/Set/Relation}
+    {Enderton.Set.Chapter\_3.exercise\_3\_23\_ii}
+
   Let $I_A$ be the identity function on the set $A$.
   That is, $I_A = \{\left< u, u \right> \mid u \in A\}$.
   Let $B$ and $C$ be any sets.
@@ -4433,8 +4460,8 @@ Show that for any sets $B$ and $C$,
 
       Let $\left< x, y \right> \in B \circ I_A$.
       By definition of the \nameref{ref:composition} of sets,
-        there exists some $t$ such that $xBt$ and $t(I_A)y$.
+        there exists some $t$ such that $x(I_A)t$ and $tBy$.
-      By definition of the identity function, $I_A(t) = y$ implies $t = y$.
+      By definition of the identity function, $I_A(x) = t$ implies $x = t$.
       Thus $xBy$.
       By hypothesis, $x \in \dom{(B \circ I_A)}$.
       Therefore $x \in \dom{I_A} = A$.

Bookshelf/Enderton/Set/Chapter_3.lean

@@ -1296,6 +1296,153 @@ theorem exercise_3_19_x
 
 end Exercise_3_19
 
+/-- #### Exercise 3.20
+
+Show that `F ↾ A = F ∩ (A × ran F)`.
+-/
+theorem exercise_3_20 {F : Set.Relation α} {A : Set α}
+  : F.restriction A = F ∩ (Set.prod A (ran F)) := by
+  calc F.restriction A
+    _ = {p | p ∈ F ∧ p.fst ∈ A} := rfl
+    _ = {p | p ∈ F ∧ p.fst ∈ A ∧ p.snd ∈ ran F} := by
+      ext x
+      have (a, b) := x
+      simp only [
+        Set.mem_setOf_eq, Set.sep_and, Set.mem_inter_iff, iff_self_and, and_imp
+      ]
+      intro hF _
+      exact ⟨hF, mem_pair_imp_snd_mem_ran hF⟩
+    _ = {p | p ∈ F} ∩ {p | p.fst ∈ A ∧ p.snd ∈ ran F} := rfl
+    _ = F ∩ {p | p.fst ∈ A ∧ p.snd ∈ ran F} := rfl
+    _ = F ∩ (Set.prod A (ran F)) := rfl
+
+/-- #### Exercise 3.22 (a)
+
+Show that the following is correct for any sets.
+```
+A ⊆ B → F⟦A⟧ ⊆ F⟦B⟧
+```
+-/
+theorem exercise_3_22_a {A B : Set α} {F : Set.Relation α} (h : A ⊆ B)
+  : F.image A ⊆ F.image B := by
+  show ∀ x, x ∈ F.image A → x ∈ F.image B
+  unfold image
+  simp only [Set.mem_setOf_eq]
+  intro x hx
+  have ⟨u, hu⟩ := hx
+  have := h hu.left
+  exact ⟨u, this, hu.right⟩
+
+/-- #### Exercise 3.22 (b)
+
+Show that the following is correct for any sets.
+```
+(F ∘ G)⟦A⟧ = F⟦G⟦A⟧⟧
+```
+-/
+theorem exercise_3_22_b {A B : Set α} {F : Set.Relation α}
+  : (F.comp G).image A = F.image (G.image A) := by
+  calc (F.comp G).image A
+    _ = { v | ∃ u ∈ A, (u, v) ∈ F.comp G } := rfl
+    _ = { v | ∃ u ∈ A, ∃ a, (u, a) ∈ G ∧ (a, v) ∈ F } := rfl
+    _ = { v | ∃ a, ∃ u ∈ A, (u, a) ∈ G ∧ (a, v) ∈ F } := by
+      ext p
+      simp only [Set.mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨u, hu, a, ha⟩
+        exact ⟨a, u, hu, ha⟩
+      · intro ⟨a, u, hu, ha⟩
+        exact ⟨u, hu, a, ha⟩
+    _ = { v | ∃ a, (∃ u ∈ A, (u, a) ∈ G) ∧ (a, v) ∈ F } := by
+      ext p
+      simp only [Set.mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨a, u, h⟩
+        exact ⟨a, ⟨u, h.left, h.right.left⟩, h.right.right⟩
+      · intro ⟨a, ⟨u, hu⟩, ha⟩
+        exact ⟨a, u, hu.left, hu.right, ha⟩
+    _ = { v | ∃ a ∈ { w | ∃ u ∈ A, (u, w) ∈ G }, (a, v) ∈ F } := rfl
+    _ = { v | ∃ a ∈ G.image A, (a, v) ∈ F } := rfl
+    _ = F.image (G.image A) := rfl
+
+/-- #### Exercise 3.22 (c)
+
+Show that the following is correct for any sets.
+```
+Q ↾ (A ∪ B) = (Q ↾ A) ∪ (Q ↾ B)
+```
+-/
+theorem exercise_3_22_c {A B : Set α} {Q : Set.Relation α}
+  : Q.restriction (A ∪ B) = (Q.restriction A) ∪ (Q.restriction B) := by
+  calc Q.restriction (A ∪ B)
+    _ = { p | p ∈ Q ∧ p.1 ∈ A ∪ B } := rfl
+    _ = { p | p ∈ Q ∧ (p.1 ∈ A ∨ p.1 ∈ B) } := rfl
+    _ = { p | (p ∈ Q ∧ p.1 ∈ A) ∨ (p ∈ Q ∧ p.1 ∈ B) } := by
+      ext p
+      simp only [Set.sep_or, Set.mem_union, Set.mem_setOf_eq]
+    _ = { p | p ∈ Q ∧ p.1 ∈ A} ∪ { p | p ∈ Q ∧ p.1 ∈ B } := rfl
+    _ = (Q.restriction A) ∪ (Q.restriction B) := rfl
+
+/-- #### Exercise 3.23 (i)
+
+Let `I` be the identity function on the set `A`. Show that for any sets `B` and
+`C`, `B ∘ I = B ↾ A`.
+-/
+theorem exercise_3_23_i {A : Set α} {B : Set.Relation α} {I : Set.Relation α}
+  (hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
+  : B.comp I = B.restriction A := by
+  rw [Set.Subset.antisymm_iff]
+  apply And.intro
+  · show ∀ p, p ∈ B.comp I → p ∈ B.restriction A
+    intro (x, y) hp
+    have ⟨t, ht⟩ := hp
+    rw [hI] at ht
+    simp only [Set.mem_setOf_eq] at ht
+    show (x, y) ∈ B ∧ x ∈ A
+    rw [← ht.left.right] at ht
+    exact ⟨ht.right, ht.left.left⟩
+  · show ∀ p, p ∈ B.restriction A → p ∈ B.comp I
+    unfold restriction comp
+    rw [hI]
+    simp only [Set.mem_setOf_eq, and_true]
+    intro (x, y) hp
+    refine ⟨x, ⟨hp.right, rfl⟩, hp.left⟩
+
+/-- #### Exercise 3.23 (ii)
+
+Let `I` be the identity function on the set `A`. Show that for any sets `B` and
+`C`, `I⟦C⟧ = A ∩ C`.
+-/
+theorem exercise_3_23_ii {A C : Set α} {I : Set.Relation α}
+  (hI : I = { p | p.1 ∈ A ∧ p.1 = p.2 })
+  : I.image C = A ∩ C := by
+  calc I.image C
+    _ = { v | ∃ u ∈ C, (u, v) ∈ I } := rfl
+    _ = { v | ∃ u ∈ C, u ∈ A ∧ u = v } := by
+      ext v
+      simp only [Set.mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨u, h₁, h₂⟩
+        rw [hI] at h₂
+        simp only [Set.mem_setOf_eq] at h₂
+        exact ⟨u, h₁, h₂⟩
+      · intro ⟨u, h₁, h₂⟩
+        refine ⟨u, h₁, ?_⟩
+        · rw [hI]
+          simp only [Set.mem_setOf_eq]
+          exact h₂
+    _ = { v | v ∈ C ∧ v ∈ A } := by
+      ext v
+      simp only [Set.mem_setOf_eq, Set.sep_mem_eq, Set.mem_inter_iff]
+      apply Iff.intro
+      · intro ⟨u, hC, hA, hv⟩
+        rw [← hv]
+        exact ⟨hC, hA⟩
+      · intro ⟨hC, hA⟩
+        exact ⟨v, hC, hA, rfl⟩
+    _ = C ∩ A := rfl
+    _ = A ∩ C := Set.inter_comm C A
+
 end Relation
 
 end Enderton.Set.Chapter_3

Bookshelf/Enderton/Set/Relation.lean

@@ -278,6 +278,8 @@ theorem single_valued_subset {F G : Set.Relation α}
   intro y₁ hy₁
   exact single_valued_eq_unique hG (h hy₁.right) (h hy)
 
+/-! ## Injections -/
+
 /--
 A `Relation` `R` is one-to-one if it is a single-rooted function.
 -/
@@ -300,6 +302,22 @@ theorem one_to_one_self_iff_one_to_one_inv {R : Relation α}
   · intro ⟨hx, hy⟩
     exact ⟨hy, hx⟩
 
+/-! ## Surjections -/
+
+/--
+Indicates `Relation` `F` is a function from `A` to `B`.
+
+This is usually denoted as `F : A → B`.
+-/
+def mapsInto (F : Relation α) (A B : Set α) :=
+  F.isSingleValued ∧ dom F = A ∧ ran F ⊆ B
+
+/--
+Indicates `Relation` `F` is a function from `A` to `ran F = B`.
+-/
+def mapsOnto (F : Relation α) (A B : Set α) :=
+  F.isSingleValued ∧ dom F = A ∧ ran F = B
+
 /-! ## Composition -/
 
 /--
@@ -341,6 +359,41 @@ theorem ran_comp_imp_ran_self {F G : Relation α}
   intro x t ht
   exact ⟨t, ht.right⟩
 
+/--
+Composition of functions is associative.
+-/
+theorem comp_assoc {R S T : Relation α}
+  : (R.comp S).comp T = R.comp (S.comp T) := by
+  calc (R.comp S).comp T
+    _ = { p | ∃ t, (p.1, t) ∈ T ∧ (t, p.2) ∈ R.comp S} := rfl
+    _ = { p | ∃ t, (p.1, t) ∈ T ∧ (∃ a, (t, a) ∈ S ∧ (a, p.2) ∈ R) } := rfl
+    _ = { p | ∃ t, ∃ a, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
+      ext p
+      simp only [mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨t, ht, a, ha⟩
+        exact ⟨t, a, ⟨ht, ha.left⟩, ha.right⟩
+      · intro ⟨t, a, h₁, h₂⟩
+        exact ⟨t, h₁.left, a, h₁.right, h₂⟩
+    _ = { p | ∃ a, ∃ t, ((p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
+      ext p
+      simp only [mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨t, a, h⟩
+        exact ⟨a, t, h⟩
+      · intro ⟨a, t, h⟩
+        exact ⟨t, a, h⟩
+    _ = { p | ∃ a, (∃ t, (p.1, t) ∈ T ∧ (t, a) ∈ S) ∧ (a, p.2) ∈ R } := by
+      ext p
+      simp only [mem_setOf_eq]
+      apply Iff.intro
+      · intro ⟨a, t, h⟩
+        exact ⟨a, ⟨t, h.left⟩, h.right⟩
+      · intro ⟨a, ⟨t, ht⟩, ha⟩
+        exact ⟨a, t, ht, ha⟩
+    _ = { p | ∃ a, (p.1, a) ∈ S.comp T ∧ (a, p.2) ∈ R } := rfl
+    _ = R.comp (S.comp T) := rfl
+
 /--
 The composition of two functions is itself a function.
 -/
@@ -420,20 +473,6 @@ theorem comp_inv_eq_inv_comp_inv {F G : Relation α}
       refine ⟨t, ?_, ?_⟩ <;> rwa [← mem_self_comm_mem_inv]
   _ = comp (inv G) (inv F) := rfl
 
-/--
-Indicates `Relation` `F` is a function from `A` to `B`.
-
-This is usually denoted as `F : A → B`.
--/
-def mapsInto (F : Relation α) (A B : Set α) :=
-  F.isSingleValued ∧ dom F = A ∧ ran F ⊆ B
-
-/--
-Indicates `Relation` `F` is a function from `A` to `ran F = B`.
--/
-def mapsOnto (F : Relation α) (A B : Set α) :=
-  F.isSingleValued ∧ dom F = A ∧ ran F = B
-
 /-! ## Ordered Pairs -/
 
 /--