Enderton. Move Kuratowski's definition of an ordered pair into chapter; not
likely to be used outside this book.finite-set-exercises
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@ -2244,7 +2244,8 @@ If not, then under what conditions does equality hold?
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\begin{proof}
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\lean{Common/Set/OrderedPair}{Set.OrderedPair.ext\_iff}
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\lean{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.OrderedPair.ext\_iff}
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Let $x$, $y$, $u$, and $v$ be arbitrary sets.
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@ -2518,7 +2519,7 @@ Show that there is no set to which every ordered pair belongs.
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\end{proof}
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\subsection{\partial{Exercise 5.5a}}%
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\subsection{\unverified{Exercise 5.5a}}%
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\label{sub:exercise-5.5a}
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Assume that $A$ and $B$ are given sets, and show that there exists a set $C$
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@ -2581,32 +2582,54 @@ In other words, show that $\{\{x\} \times B \mid x \in A\}$ is a set.
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\end{proof}
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\subsection{\partial{Exercise 5.5b}}%
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\subsection{\verified{Exercise 5.5b}}%
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\label{sub:exercise-5.5b}
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With $A$, $B$, and $C$ as above, show that $A \times B = \bigcup C$.
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\begin{proof}
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\lean{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.exercise\_5\_5b}
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Let $A$ and $B$ be arbitrary sets.
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We want to show that
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\begin{equation}
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\label{sub:exercise-5.5b-eq1}
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A \times B = \bigcup\; \{\{x\} \times B \mid x \in A\}.
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\end{equation}
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Note that \nameref{sub:cartesian-product} and \nameref{sub:exercise-5.5a}
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prove the left- and right-hand sides of \eqref{sub:exercise-5.5b-eq1} are
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sets respectively.
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Then
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\begin{align*}
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A \times B
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& = \{ y \mid \exists x \in A, \exists b \in B, y = \left< x, b \right> \} \\
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& = \{ y \mid \exists b \in B, \exists x \in A, y = \left< x, b \right> \} \\
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& = \{ y \mid \exists b \in B, y \in \{ \left< x, b \right> \mid x \in A \} \} \\
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& = \{ y \mid y \in \{ \left< x, b \right> \mid x \in A \land b \in B \} \} \\
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& = \{ y \mid \exists z \in \{\{x\} \times B \mid x \in A \}, y \in z \} \\
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& = \bigcup \{\{x\} \times B \mid x \in A \}.
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\end{align*}
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The left-hand side of \eqref{sub:exercise-5.5b-eq1} is a set by virtue of
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\nameref{sub:cartesian-product}.
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The right-hand side of \eqref{sub:exercise-5.5b-eq1} is a set by virtue of
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\nameref{sub:exercise-5.5a}.
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We prove the set on each side is a subset of the other.
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\paragraph{($\subseteq$)}%
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Let $c \in A \times B$.
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Then there exists some $a \in A$ and $b \in B$ such that
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$c = \left< a, b \right>$.
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Thus $c \in \{a\} \times B$.
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We also note $\{a\} \times B \in \{\{x\} \times B \mid x \in A\}$,
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specifically when $x = a$.
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Therefore, by the \nameref{ref:union-axiom},
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$c \in \bigcup\;\{\{x\} \times B \mid x \in A\}$.
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\paragraph{($\supseteq$)}%
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Let $c \in \bigcup\; \{\{x\} \times B \mid x \in A\}$.
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By the \nameref{ref:union-axiom}, there exists some
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$b \in \{\{x\} \times B \mid x \in A\}$ such that $c \in b$.
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Then there exists some $x \in A$ such that $b = \{x\} \times B$.
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Therefore $c \in \{x\} \times B$.
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But $x \in A$ meaning $c \in A \times B$ as well.
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\paragraph{Conclusion}%
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Since we have shown
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$A \times B \subseteq \bigcup\; \{\{x\} \times B \mid x \in A\}$ and
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$A \times B \supseteq \bigcup\; \{\{x\} \times B \mid x \in A\}$, it
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follows \eqref{sub:exercise-5.5b-eq1} is a true identity.
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\end{proof}
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@ -1,8 +1,7 @@
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import Mathlib.Data.Set.Basic
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import Mathlib.SetTheory.ZFC.Basic
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import Common.Logic.Basic
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import Common.Set.Basic
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import Common.Set.OrderedPair
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/-! # Enderton.Chapter_3
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@ -11,13 +10,107 @@ Relations and Functions
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namespace Enderton.Set.Chapter_3
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/-! ## Ordered Pairs -/
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/--
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Kazimierz Kuratowski's definition of an ordered pair.
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-/
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def OrderedPair (x : α) (y : β) : Set (Set (α ⊕ β)) :=
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{{Sum.inl x}, {Sum.inl x, Sum.inr y}}
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namespace OrderedPair
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/--
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For any sets `x`, `y`, `u`, and `v`, `⟨u, v⟩ = ⟨x, y⟩` **iff** `u = x ∧ v = y`.
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-/
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theorem ext_iff {x u : α} {y v : β}
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: (OrderedPair x y = OrderedPair u v) ↔ (x = u ∧ y = v) := by
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unfold OrderedPair
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apply Iff.intro
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· intro h
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have hu := Set.ext_iff.mp h {Sum.inl u}
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have huv := Set.ext_iff.mp h {Sum.inl u, Sum.inr v}
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simp only [
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Set.mem_singleton_iff,
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Set.mem_insert_iff,
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true_or,
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iff_true
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] at hu
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simp only [
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Set.mem_singleton_iff,
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Set.mem_insert_iff,
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or_true,
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iff_true
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] at huv
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apply Or.elim hu
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· apply Or.elim huv
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· -- #### Case 1
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-- `{u} = {x}` and `{u, v} = {x}`.
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intro huv_x hu_x
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rw [Set.singleton_eq_singleton_iff] at hu_x
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rw [hu_x] at huv_x
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have hx_v := Set.pair_eq_singleton_mem_imp_eq_self huv_x
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rw [hu_x, hx_v] at h
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simp only [Set.mem_singleton_iff, Set.insert_eq_of_mem] at h
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have := Set.pair_eq_singleton_mem_imp_eq_self $
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Set.pair_eq_singleton_mem_imp_eq_self h
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rw [← hx_v] at this
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injection hu_x with p
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injection this with q
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exact ⟨p.symm, q⟩
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· -- #### Case 2
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-- `{u} = {x}` and `{u, v} = {x, y}`.
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intro huv_xy hu_x
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rw [Set.singleton_eq_singleton_iff] at hu_x
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rw [hu_x] at huv_xy
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by_cases hx_v : Sum.inl x = Sum.inr v
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· rw [hx_v] at huv_xy
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simp at huv_xy
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have := Set.pair_eq_singleton_mem_imp_eq_self huv_xy.symm
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injection hu_x with p
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injection this with q
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exact ⟨p.symm, q⟩
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· rw [Set.ext_iff] at huv_xy
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have := huv_xy (Sum.inr v)
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simp at this
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injection hu_x with p
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exact ⟨p.symm, this.symm⟩
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· apply Or.elim huv
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· -- #### Case 3
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-- `{u} = {x, y}` and `{u, v} = {x}`.
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intro huv_x _
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rw [Set.ext_iff] at huv_x
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have hv_x := huv_x (Sum.inr v)
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simp only [
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Set.mem_singleton_iff,
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Set.mem_insert_iff,
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or_true,
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true_iff
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] at hv_x
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· -- #### Case 4
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-- `{u} = {x, y}` and `{u, v} = {x, y}`.
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intro _ hu_xy
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rw [Set.ext_iff] at hu_xy
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have hy_u := hu_xy (Sum.inr y)
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simp only [
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Set.mem_singleton_iff,
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Set.mem_insert_iff,
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or_true,
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iff_true
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] at hy_u
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· intro h
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rw [h.left, h.right]
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end OrderedPair
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/-- ### Theorem 3B
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If `x ∈ C` and `y ∈ C`, then `⟨x, y⟩ ∈ 𝒫 𝒫 C`.
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-/
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theorem theorem_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
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: Set.OrderedPair x y ∈ 𝒫 𝒫 C := by
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have hxs : {x} ⊆ C := Set.singleton_subset_iff.mpr hx
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have hxys : {x, y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
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theorem theorem_3b {C : Set (α ⊕ α)} (hx : Sum.inl x ∈ C) (hy : Sum.inr y ∈ C)
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: OrderedPair x y ∈ 𝒫 𝒫 C := by
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have hxs : {Sum.inl x} ⊆ C := Set.singleton_subset_iff.mpr hx
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have hxys : {Sum.inl x, Sum.inr y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
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exact Set.mem_mem_imp_pair_subset hxs hxys
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/-- ### Exercise 5.1
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@ -25,7 +118,8 @@ theorem theorem_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
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Suppose that we attempted to generalize the Kuratowski definitions of ordered
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pairs to ordered triples by defining
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```
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⟨x, y, z⟩* = {{x}, {x, y}, {x, y, z}}.
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⟨x, y, z⟩* = {{x}, {x, y}, {x, y, z}}.open Set
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```
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Show that this definition is unsuccessful by giving examples of objects `u`,
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`v`, `w`, `x`, `y`, `z` with `⟨x, y, z⟩* = ⟨u, v, w⟩*` but with either `y ≠ v`
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@ -115,4 +209,53 @@ theorem exercise_5_3 {A : Set (Set α)} {𝓑 : Set (Set β)}
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· intro ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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exact ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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/-- ### Exercise 5.5a
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Assume that `A` and `B` are given sets, and show that there exists a set `C`
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such that for any `y`,
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```
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y ∈ C ↔ y = {x} × B for some x in A.
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```
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In other words, show that `{{x} × B | x ∈ A}` is a set.
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-/
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theorem exercise_5_5a {A : Set α} {B : Set β}
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: ∃ C : Set (Set (α × β)),
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y ∈ C ↔ ∃ x ∈ A, y = Set.prod {x} B := by
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sorry
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/-- ### Exercise 5.5b
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With `A`, `B`, and `C` as above, show that `A × B = ∪ C`.
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-/
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theorem exercise_5_5b {A : Set α} (B : Set β)
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: Set.prod A B = ⋃₀ {Set.prod ({x} : Set α) B | x ∈ A} := by
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suffices Set.prod A B ⊆ ⋃₀ {Set.prod {x} B | x ∈ A} ∧
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⋃₀ {Set.prod {x} B | x ∈ A} ⊆ Set.prod A B from
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Set.Subset.antisymm_iff.mpr this
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apply And.intro
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· show ∀ t, t ∈ Set.prod A B → t ∈ ⋃₀ {Set.prod {x} B | x ∈ A}
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intro t h
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simp only [Set.mem_setOf_eq] at h
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unfold Set.sUnion sSup Set.instSupSetSet
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simp only [Set.mem_setOf_eq, exists_exists_and_eq_and]
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unfold Set.prod at h
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simp only [Set.mem_setOf_eq] at h
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refine ⟨t.fst, ⟨h.left, ?_⟩⟩
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unfold Set.prod
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq, true_and]
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exact h.right
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· show ∀ t, t ∈ ⋃₀ {Set.prod {x} B | x ∈ A} → t ∈ Set.prod A B
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unfold Set.prod
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intro t ht
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simp only [
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Set.mem_singleton_iff,
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Set.mem_sUnion,
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Set.mem_setOf_eq,
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exists_exists_and_eq_and
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] at ht
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have ⟨a, ⟨h, ⟨ha, hb⟩⟩⟩ := ht
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simp only [Set.mem_setOf_eq]
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rw [← ha] at h
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exact ⟨h, hb⟩
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end Enderton.Set.Chapter_3
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@ -1,4 +1,3 @@
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import Common.Set.Basic
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import Common.Set.Interval
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import Common.Set.OrderedPair
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import Common.Set.Partition
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@ -1,98 +0,0 @@
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import Mathlib.Data.Set.Basic
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import Common.Logic.Basic
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import Common.Set.Basic
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namespace Set
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/--
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Kazimierz Kuratowski's definition of an ordered pair.
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Like `Set`, this is a homogeneous structure.
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-/
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def OrderedPair (x y : α) : Set (Set α) := {{x}, {x, y}}
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namespace OrderedPair
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/--
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For any sets `x`, `y`, `u`, and `v`, `⟨u, v⟩ = ⟨x, y⟩` **iff** `u = x ∧ v = y`.
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-/
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theorem ext_iff
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: (OrderedPair x y = OrderedPair u v) ↔ (x = u ∧ y = v) := by
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unfold OrderedPair
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apply Iff.intro
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· intro h
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have h' := h
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rw [Set.ext_iff] at h'
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have hu := h' {u}
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have huv := h' {u, v}
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simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hu
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simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at huv
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apply Or.elim hu
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· apply Or.elim huv
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· -- #### Case 1
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-- `{u} = {x}` and `{u, v} = {x}`.
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intro huv_x hu_x
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rw [singleton_eq_singleton_iff] at hu_x
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rw [hu_x] at huv_x
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have hx_v := pair_eq_singleton_mem_imp_eq_self huv_x
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rw [hu_x, hx_v] at h
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simp only [mem_singleton_iff, insert_eq_of_mem] at h
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have := pair_eq_singleton_mem_imp_eq_self $
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pair_eq_singleton_mem_imp_eq_self h
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rw [← hx_v] at this
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exact ⟨hu_x.symm, this⟩
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· -- #### Case 2
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-- `{u} = {x}` and `{u, v} = {x, y}`.
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intro huv_xy hu_x
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rw [singleton_eq_singleton_iff] at hu_x
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rw [hu_x] at huv_xy
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by_cases hx_v : x = v
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· rw [hx_v] at huv_xy
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simp at huv_xy
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have := pair_eq_singleton_mem_imp_eq_self huv_xy.symm
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exact ⟨hu_x.symm, this⟩
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· rw [Set.ext_iff] at huv_xy
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have := huv_xy v
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simp at this
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apply Or.elim this
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· intro hv_x
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rw [hu_x, ← hv_x] at h
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simp at h
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have := pair_eq_singleton_mem_imp_eq_self $
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pair_eq_singleton_mem_imp_eq_self h
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exact ⟨hu_x.symm, this⟩
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· intro hv_y
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exact ⟨hu_x.symm, hv_y.symm⟩
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· apply Or.elim huv
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· -- #### Case 3
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-- `{u} = {x, y}` and `{u, v} = {x}`.
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intro huv_x hu_xy
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rw [Set.ext_iff] at huv_x
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have hu_x := huv_x u
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have hv_x := huv_x v
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simp only [mem_singleton_iff, mem_insert_iff, true_or, true_iff] at hu_x
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simp only [mem_singleton_iff, mem_insert_iff, or_true, true_iff] at hv_x
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rw [← hu_x] at hu_xy
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have := pair_eq_singleton_mem_imp_eq_self hu_xy.symm
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rw [hu_x, ← hv_x] at this
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exact ⟨hu_x.symm, this⟩
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· -- #### Case 4
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-- `{u} = {x, y}` and `{u, v} = {x, y}`.
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intro huv_xy hu_xy
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rw [Set.ext_iff] at hu_xy
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have hx_u := hu_xy x
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have hy_u := hu_xy y
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simp only [mem_singleton_iff, mem_insert_iff, true_or, iff_true] at hx_u
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simp only [mem_singleton_iff, mem_insert_iff, or_true, iff_true] at hy_u
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rw [hx_u, hy_u] at huv_xy
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simp only [mem_singleton_iff, insert_eq_of_mem] at huv_xy
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have := pair_eq_singleton_mem_imp_eq_self huv_xy
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rw [← this] at hy_u
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exact ⟨hx_u, hy_u⟩
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· intro h
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rw [h.left, h.right]
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end OrderedPair
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end Set
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