Enderton. Add additional definitions and prove Theorem 3E.
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@ -24,12 +24,18 @@
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\chapter{Reference}%
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\label{chap:reference}
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\section{\partial{Composition}}%
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\section{\defined{Composition}}%
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\label{ref:composition}
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The \textbf{composition} of sets $F$ and $G$ is
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$$F \circ G = \{\left< u, v \right> \mid \exists t(uGt \land tFv)\}.$$
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\begin{definition}
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\lean*{Common/Set/Relation}{Set.Relation.comp}
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\end{definition}
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\section{\defined{Domain}}%
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\label{ref:domain}
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@ -105,7 +111,7 @@ One-to-one functions are sometimes called \textbf{injections}.
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\end{definition}
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\section{\partial{Image}}%
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\section{\defined{Image}}%
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\label{ref:image}
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Let $A$ and $F$ be arbitrary sets.
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@ -116,12 +122,24 @@ The \textbf{image of $A$ under $F$} is the set
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& = \{v \mid (\exists u \in A) uFv\}.
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\end{align*}
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\section{\partial{Inverse}}%
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\begin{definition}
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\lean*{Common/Set/Relation}{Set.Relation.image}
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\end{definition}
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\section{\defined{Inverse}}%
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\label{ref:inverse}
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The \textbf{inverse} of a set $F$ is the set
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$$F^{-1} = \{\left< u, v \right> \mid vFu\}.$$
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\begin{definition}
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\lean*{Common/Set/Relation}{Set.Relation.inv}
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\end{definition}
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\section{\defined{Ordered Pair}}%
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\label{ref:ordered-pair}
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@ -214,12 +232,18 @@ A \textbf{relation} is a set of \nameref{ref:ordered-pair}s.
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\end{definition}
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\section{\partial{Restriction}}%
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\section{\defined{Restriction}}%
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\label{ref:restriction}
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The \textbf{restriction} of a set $F$ to set $A$ is the set
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$$F \restriction A = \{\left< u, v \right> \mid uFv \land u \in A\}.$$
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\begin{definition}
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\lean*{Common/Set/Relation}{Set.Relation.restriction}
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\end{definition}
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\section{\defined{Subset Axioms}}%
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\label{ref:subset-axioms}
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@ -2997,7 +3021,7 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
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\section{Functions}%
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\label{sec:functions}
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\subsection{\partial{Theorem 3E}}%
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\subsection{\verified{Theorem 3E}}%
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\label{sub:theorem-3e}
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\begin{theorem}[3E]
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@ -3009,6 +3033,17 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
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\begin{proof}
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\statementpadding
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.theorem\_3e\_i}
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.theorem\_3e\_ii}
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\lean{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.theorem\_3e\_iii}
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We prove that (i) $\dom{(F^{-1})} = \ran{F}$, (ii) $\ran{(F^{-1})} = \dom{F}$,
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and (iii) $(F^{-1})^{-1} = F$.
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@ -3056,6 +3091,14 @@ Show that an ordered $4$-tuple is also an ordered $m$-tuple for every positive
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\begin{proof}
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\statementpadding
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\lean*{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.theorem\_3f\_i}
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\lean{Bookshelf/Enderton/Set/Chapter\_3}
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{Enderton.Set.Chapter\_3.theorem\_3f\_ii}
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TODO
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\end{proof}
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@ -100,20 +100,20 @@ theorem exercise_5_3 {A : Set (Set α)} {𝓑 : Set (Set β)}
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ext x
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rw [Set.mem_setOf_eq]
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apply Iff.intro
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· intro ⟨h₁, ⟨b, h₂⟩⟩
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exact ⟨b, ⟨h₂.left, ⟨h₁, h₂.right⟩⟩⟩
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· intro ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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exact ⟨h₂, ⟨b, ⟨h₁, h₃⟩⟩⟩
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· intro ⟨h₁, b, h₂⟩
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exact ⟨b, h₂.left, h₁, h₂.right⟩
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· intro ⟨b, h₁, h₂, h₃⟩
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exact ⟨h₂, b, h₁, h₃⟩
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_ = ⋃₀ { Set.prod A p | p ∈ 𝓑 } := by
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ext x
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rw [Set.mem_setOf_eq]
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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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apply Iff.intro
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· intro ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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exact ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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· intro ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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exact ⟨b, ⟨h₁, ⟨h₂, h₃⟩⟩⟩
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· intro ⟨b, h₁, h₂, h₃⟩
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exact ⟨b, h₁, h₂, h₃⟩
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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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@ -268,7 +268,7 @@ theorem exercise_6_7 {R : Set.Relation α}
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Set.mem_image, Prod.exists, exists_and_right, exists_eq_right
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] at hd
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have ⟨y, hp⟩ := hd
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have hm : OrderedPair x y ∈ R.image (fun p => OrderedPair p.1 p.2) := by
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have hm : OrderedPair x y ∈ Set.image (fun p => OrderedPair p.1 p.2) R := by
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unfold Set.image
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simp only [Prod.exists, Set.mem_setOf_eq]
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exact ⟨x, ⟨y, ⟨hp, rfl⟩⟩⟩
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@ -279,7 +279,7 @@ theorem exercise_6_7 {R : Set.Relation α}
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unfold Set.Relation.ran Prod.snd at hr
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simp only [Set.mem_image, Prod.exists, exists_eq_right] at hr
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have ⟨t, ht⟩ := hr
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have hm : OrderedPair t x ∈ R.image (fun p => OrderedPair p.1 p.2) := by
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have hm : OrderedPair t x ∈ Set.image (fun p => OrderedPair p.1 p.2) R := by
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simp only [Set.mem_image, Prod.exists]
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exact ⟨t, ⟨x, ⟨ht, rfl⟩⟩⟩
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unfold OrderedPair at hm
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@ -329,7 +329,7 @@ theorem exercise_6_7 {R : Set.Relation α}
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simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
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exact hxy_mem this
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/-- ### Exercise 6.8i
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/-- ### Exercise 6.8 (i)
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Show that for any set `𝓐`:
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```
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@ -350,12 +350,12 @@ theorem exercise_6_8_i {A : Set (Set.Relation α)}
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exists_exists_and_eq_and
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]
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apply Iff.intro
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· intro ⟨y, ⟨t, ⟨ht, hx⟩⟩⟩
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exact ⟨t, ⟨ht, ⟨y, hx⟩⟩⟩
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· intro ⟨t, ⟨ht, ⟨y, hx⟩⟩⟩
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exact ⟨y, ⟨t, ⟨ht, hx⟩⟩⟩
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· intro ⟨y, t, ht, hx⟩
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exact ⟨t, ht, y, hx⟩
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· intro ⟨t, ht, y, hx⟩
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exact ⟨y, t, ht, hx⟩
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/-- ### Exercise 6.8ii
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/-- ### Exercise 6.8 (ii)
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Show that for any set `𝓐`:
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```
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@ -380,7 +380,7 @@ theorem exercise_6_8_ii {A : Set (Set.Relation α)}
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· intro ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
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exact ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
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/-- ## Exercise 6.9i
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/-- ### Exercise 6.9 (i)
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Discuss the result of replacing the union operation by the intersection
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operation in the preceding problem.
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@ -406,7 +406,7 @@ theorem exercise_6_9_i {A : Set (Set.Relation α)}
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intro _ y hy R hR
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exact ⟨y, hy R hR⟩
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/-- ## Exercise 6.9ii
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/-- ### Exercise 6.9 (ii)
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Discuss the result of replacing the union operation by the intersection
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operation in the preceding problem.
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@ -432,4 +432,86 @@ theorem exercise_6_9_ii {A : Set (Set.Relation α)}
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intro _ y hy R hR
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exact ⟨y, hy R hR⟩
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/-- ### Theorem 3E (i)
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For a set `F`, `dom F⁻¹ = ran F`.
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-/
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theorem theorem_3e_i {F : Set.Relation α}
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: Set.Relation.dom (F.inv) = Set.Relation.ran F := by
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ext x
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unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
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simp only [
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Prod.exists,
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Set.mem_image,
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Set.mem_setOf_eq,
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Prod.mk.injEq,
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exists_and_right,
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exists_eq_right
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]
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apply Iff.intro
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· intro ⟨y, a, _, h⟩
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rw [← h.right.left]
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exact ⟨a, h.left⟩
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· intro ⟨y, hy⟩
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exact ⟨y, y, x, hy, rfl, rfl⟩
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/-- ### Theorem 3E (ii)
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For a set `F`, `ran F⁻¹ = dom F`.
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-/
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theorem theorem_3e_ii {F : Set.Relation α}
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: Set.Relation.ran (F.inv) = Set.Relation.dom F := by
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ext x
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unfold Set.Relation.dom Set.Relation.ran Set.Relation.inv
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simp only [
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Prod.exists,
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Set.mem_image,
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Set.mem_setOf_eq,
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Prod.mk.injEq,
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exists_eq_right,
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exists_and_right
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]
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apply Iff.intro
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· intro ⟨a, y, b, h⟩
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rw [← h.right.right]
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exact ⟨b, h.left⟩
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· intro ⟨y, hy⟩
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exact ⟨y, x, y, hy, rfl, rfl⟩
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/-- ### Theorem 3E (iii)
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For a set `F`, `(F⁻¹)⁻¹ = F`.
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-/
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theorem theorem_3e_iii {F : Set.Relation α}
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: F.inv.inv = F := by
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unfold Set.Relation.inv
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simp only [Prod.exists, Set.mem_setOf_eq, Prod.mk.injEq]
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ext x
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apply Iff.intro
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· intro hx
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have ⟨a₁, b₁, ⟨⟨a₂, b₂, h₁⟩, h₂⟩⟩ := hx
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rw [← h₂, ← h₁.right.right, ← h₁.right.left]
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exact h₁.left
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· intro hx
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have (p, q) := x
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refine ⟨q, p, ⟨?_, ?_⟩⟩
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· exact ⟨p, q, hx, rfl, rfl⟩
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· rfl
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/-- ### Theorem 3F (i)
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For a set `F`, `F⁻¹` is a function **iff** `F` is single-rooted.
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-/
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theorem theorem_3f_i {F : Set.Relation α}
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: Set.Relation.isSingleValued F.inv ↔ Set.Relation.isSingleRooted F := by
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sorry
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/-- ### Theorem 3F (ii)
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For a relation `F`, `F` is a function **iff** `F⁻¹` is single-rooted.
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-/
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theorem theorem_3f_ii {F : Set.Relation α}
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: Set.Relation.isSingleValued F ↔ Set.Relation.isSingleRooted F.inv := by
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sorry
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end Enderton.Set.Chapter_3
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@ -37,11 +37,51 @@ The field of a `Relation`.
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-/
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def fld (R : Relation α) : Set α := dom R ∪ ran R
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/--
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The inverse of a `Relation`.
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-/
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def inv (R : Relation α) : Relation α := { (p.2, p.1) | p ∈ R }
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/--
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The composition of two `Relation`s.
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-/
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def comp (F G : Relation α) : Relation α :=
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{ p | ∃ t, (p.1, t) ∈ G ∧ (t, p.2) ∈ F}
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/--
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The restriction of a `Relation` to a `Set`.
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-/
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def restriction (R : Relation α) (A : Set α) : Relation α :=
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{ p ∈ R | p.1 ∈ A }
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/--
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The image of a `Relation` under a `Set`.
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-/
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def image (R : Relation α) (A : Set α) : Set α :=
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{ y | ∃ u ∈ A, (u, y) ∈ R }
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/--
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A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, there
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exists exactly one `x` such that `⟨x, y⟩ ∈ R`.
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-/
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def isSingleRooted (R : Relation α) : Prop :=
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∀ y ∈ R.ran, ∃! x, x ∈ R.dom ∧ (x, y) ∈ R
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/--
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A `Relation` `R` is said to be single-valued **iff** for all `x ∈ dom R`, there
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exists exactly one `y` such that `⟨x, y⟩ ∈ R`.
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Notice, a `Relation` that is single-valued is a function.
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-/
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def isSingleValued (R : Relation α) : Prop :=
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∀ x ∈ R.dom, ∃! y, y ∈ R.ran ∧ (x, y) ∈ R
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/--
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Convert a `Relation` into an equivalent representation using `OrderedPair`s.
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-/
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def toOrderedPairs (R : Relation α) : Set (Set (Set α)) :=
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R.image (fun (x, y) => OrderedPair x y)
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-- Notice here we are using `Set.image` and *not* `Set.Relation.image`.
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Set.image (fun (x, y) => OrderedPair x y) R
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end Relation
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