Enderton. Finish exercises 6.

Bookshelf/Enderton/Set.tex

@@ -2792,7 +2792,7 @@ Show that if $R$ is a relation, then $\fld{R} = \bigcup\bigcup R$.
 
 \end{proof}
 
-\subsection{\unverified{Exercise 6.8}}%
+\subsection{\verified{Exercise 6.8}}%
 \label{sub:exercise-6.8}
 
 Show that for any set $\mathscr{A}$:
@@ -2807,21 +2807,112 @@ Show that for any set $\mathscr{A}$:
 
 \begin{proof}
 
-  TODO
+  \statementpadding
+
+  \lean*{Bookshelf/Enderton/Set/Chapter\_3}
+    {Enderton.Set.Chapter\_3.exercise\_6\_8\_i}
+
+  \lean{Bookshelf/Enderton/Set/Chapter\_3}
+    {Enderton.Set.Chapter\_3.exercise\_6\_8\_ii}
+
+  We prove (i) \eqref{sub:exercise-6.8-eq1} and then (ii)
+    \eqref{sub:exercise-6.8-eq2}.
+
+  \paragraph{(i)}%
+
+    Let $x \in \dom{\bigcup{\mathscr{A}}}$.
+    By definition of a domain, there exists some $y$ such that
+      $\left< x, y \right> \in \bigcup{\mathscr{A}}$.
+    By definition of the union of sets,
+      $\exists y, \exists R \in \mathscr{A}, \left< x, y \right> \in R$.
+    Equivalently,
+      $\exists R \in \mathscr{A}, \exists y, \left< x, y \right> \in R$.
+    By another application of the definition of a domain,
+      $\exists R \in \mathscr{A}, x \in \dom{R}$.
+    By another application of the definition of the union of sets,
+      $x \in \bigcup\;\{ \dom{R} \mid R \in \mathscr{A} \}$.
+    Since membership of these two sets holds biconditionally, it follows
+      \eqref{sub:exercise-6.8-eq1} holds.
+
+  \paragraph{(ii)}%
+
+    Let $x \in \ran{\bigcup{\mathscr{A}}}$.
+    By definition of a range, there exists some $t$ such that
+      $\left< t, x \right> \in \bigcup{\mathscr{A}}$.
+    By definition of the union of sets,
+      $\exists t, \exists R \in \mathscr{A}, \left< t, x \right> \in R$.
+    Equivalently,
+      $\exists R \in \mathscr{A}, \exists t, \left< t, x \right> \in R$.
+    By another application of the definition of a range,
+      $\exists R \in \mathscr{A}, x \in \ran{R}$.
+    By another application of the definition of the union of sets,
+      $x \in \bigcup\;\{ \ran{R} \mid R \in \mathscr{A} \}$.
+    Since membership of these two sets holds biconditionally, it follows
+      \eqref{sub:exercise-6.8-eq2} holds.
 
 \end{proof}
 
-\subsection{\unverified{Exercise 6.9}}%
+\subsection{\verified{Exercise 6.9}}%
 \label{sub:exercise-6.9}
 
 Discuss the result of replacing the union operation by the intersection
   operation in the preceding problem.
 
-\begin{proof}
+\begin{answer}
 
-  TODO
+  \statementpadding
 
-\end{proof}
+  \lean*{Bookshelf/Enderton/Set/Chapter\_3}
+    {Enderton.Set.Chapter\_3.exercise\_6\_9\_i}
+
+  \lean{Bookshelf/Enderton/Set/Chapter\_3}
+    {Enderton.Set.Chapter\_3.exercise\_6\_9\_ii}
+
+  Replacing the union operation with the intersection problem produces the
+    following relationships
+    \begin{align}
+      \dom{\bigcap{\mathscr{A}}}
+        & \subseteq \bigcap\;\{ \dom{R} \mid R \in \mathscr{A} \},
+        & \label{sub:exercise-6.9-eq1} \\
+      \ran{\bigcap{\mathscr{A}}}
+        & \subseteq \bigcap\;\{ \ran{R} \mid R \in \mathscr{A} \}.
+        & \label{sub:exercise-6.9-eq2}
+    \end{align}
+
+  We prove (i) \eqref{sub:exercise-6.9-eq1} and then (ii)
+    \eqref{sub:exercise-6.9-eq2}.
+
+  \paragraph{(i)}%
+
+    Let $x \in \dom{\bigcap{\mathscr{A}}}$.
+    By definition of the domain of a set,
+      $\exists y, \left< x, y \right> \in \bigcap{\mathscr{A}}$.
+    By definition of the intersection of sets,
+      $\exists y, \forall R \in \mathscr{A}, \left< x, y \right> \in R$.
+    But this implies that
+      $\forall R \in \mathscr{A}, \exists y, \left< x, y \right> \in R$.
+    By another application of the definition of the domain of a set,
+      $\forall R \in \mathscr{A}, x \in \dom{R}$.
+    By another application of the intersection of sets,
+      $x \in \bigcap\;\{ \dom{R} \mid R \in \mathscr{A} \}$.
+      Thus \eqref{sub:exercise-6.9-eq1} holds.
+
+  \paragraph{(ii)}%
+
+    Let $x \in \ran{\bigcap{\mathscr{A}}}$.
+    By definition of the range of a set,
+      $\exists t, \left< t, x \right> \in \bigcap{\mathscr{A}}$.
+    By definition of the intersection of sets,
+      $\exists t, \forall R \in \mathscr{A}, \left< t, x \right> \in R$.
+    But this implies that
+      $\forall R \in \mathscr{A}, \exists t, \left< t, x \right> \in R$.
+    By another application of the definition of the domain of a set,
+      $\forall R \in \mathscr{A}, x \in \ran{R}$.
+    By another application of the intersection of sets,
+      $x \in \bigcap\;\{ \ran{R} \mid R \in \mathscr{A} \}$.
+      Thus \eqref{sub:exercise-6.9-eq2} holds.
+
+\end{answer}
 
 \section{Exercises 7}%
 \label{sec:exercises-7}

Bookshelf/Enderton/Set/Chapter_3.lean

@@ -329,4 +329,107 @@ theorem exercise_6_7 {R : Set.Relation α}
       simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
       exact hxy_mem this
 
+/-- ### Exercise 6.8i
+
+Show that for any set `𝓐`:
+```
+dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
+```
+-/
+theorem exercise_6_8_i {A : Set (Set.Relation α)}
+  : Set.Relation.dom (⋃₀ A) = ⋃₀ { Set.Relation.dom R | R ∈ A } := by
+  ext x
+  unfold Set.Relation.dom Prod.fst
+  simp only [
+    Set.mem_image,
+    Set.mem_sUnion,
+    Prod.exists,
+    exists_and_right,
+    exists_eq_right,
+    Set.mem_setOf_eq,
+    exists_exists_and_eq_and
+  ]
+  apply Iff.intro
+  · intro ⟨y, ⟨t, ⟨ht, hx⟩⟩⟩
+    exact ⟨t, ⟨ht, ⟨y, hx⟩⟩⟩
+  · intro ⟨t, ⟨ht, ⟨y, hx⟩⟩⟩
+    exact ⟨y, ⟨t, ⟨ht, hx⟩⟩⟩
+
+/-- ### Exercise 6.8ii
+
+Show that for any set `𝓐`:
+```
+ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
+```
+-/
+theorem exercise_6_8_ii {A : Set (Set.Relation α)}
+  : Set.Relation.ran (⋃₀ A) = ⋃₀ { Set.Relation.ran R | R ∈ A } := by
+  ext x
+  unfold Set.Relation.ran Prod.snd
+  simp only [
+    Set.mem_image,
+    Set.mem_sUnion,
+    Prod.exists,
+    exists_eq_right,
+    Set.mem_setOf_eq,
+    exists_exists_and_eq_and
+  ]
+  apply Iff.intro
+  · intro ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
+    exact ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
+  · intro ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
+    exact ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
+
+/-- ## Exercise 6.9i
+
+Discuss the result of replacing the union operation by the intersection
+operation in the preceding problem.
+```
+dom ⋃ A = ⋃ { dom R | R ∈ 𝓐 }
+```
+-/
+theorem exercise_6_9_i {A : Set (Set.Relation α)}
+  : Set.Relation.dom (⋂₀ A) ⊆ ⋂₀ { Set.Relation.dom R | R ∈ A } := by
+  show ∀ x, x ∈ Set.Relation.dom (⋂₀ A) → x ∈ ⋂₀ { Set.Relation.dom R | R ∈ A }
+  unfold Set.Relation.dom Prod.fst
+  simp only [
+    Set.mem_image,
+    Set.mem_sInter,
+    Prod.exists,
+    exists_and_right,
+    exists_eq_right,
+    Set.mem_setOf_eq,
+    forall_exists_index,
+    and_imp,
+    forall_apply_eq_imp_iff₂
+  ]
+  intro _ y hy R hR
+  exact ⟨y, hy R hR⟩
+
+/-- ## Exercise 6.9ii
+
+Discuss the result of replacing the union operation by the intersection
+operation in the preceding problem.
+```
+ran ⋃ A = ⋃ { ran R | R ∈ 𝓐 }
+```
+-/
+theorem exercise_6_9_ii {A : Set (Set.Relation α)}
+  : Set.Relation.ran (⋂₀ A) ⊆ ⋂₀ { Set.Relation.ran R | R ∈ A } := by
+  show ∀ x, x ∈ Set.Relation.ran (⋂₀ A) → x ∈ ⋂₀ { Set.Relation.ran R | R ∈ A }
+  unfold Set.Relation.ran Prod.snd
+  simp only [
+    Set.mem_image,
+    Set.mem_sInter,
+    Prod.exists,
+    exists_and_right,
+    exists_eq_right,
+    Set.mem_setOf_eq,
+    forall_exists_index,
+    and_imp,
+    forall_apply_eq_imp_iff₂
+  ]
+  intro _ y hy R hR
+  exact ⟨y, hy R hR⟩
+
 end Enderton.Set.Chapter_3