Normalize header sizes.
parent
e869d6f2d3
commit
63667c22e4
|
|
@ -10,7 +10,7 @@ Axioms and Operations
|
|||
namespace Enderton.Set.Chapter_2
|
||||
|
||||
|
||||
/-- ### Exercise 2.1
|
||||
/-- #### Exercise 2.1
|
||||
|
||||
Assume that `A` is the set of integers divisible by `4`. Similarly assume that
|
||||
`B` and `C` are the sets of integers divisible by `9` and `10`, respectively.
|
||||
|
|
@ -32,7 +32,7 @@ theorem exercise_2_1 {A B C : Set ℤ}
|
|||
· rw [hC] at hc
|
||||
exact Set.mem_setOf.mp hc
|
||||
|
||||
/-- ### Exercise 2.2
|
||||
/-- #### Exercise 2.2
|
||||
|
||||
Give an example of sets `A` and `B` for which `⋃ A = ⋃ B` but `A ≠ B`.
|
||||
-/
|
||||
|
|
@ -71,7 +71,7 @@ theorem exercise_2_2 {A B : Set (Set ℕ)}
|
|||
have h₂ := h₁ 2
|
||||
simp at h₂
|
||||
|
||||
/-- ### Exercise 2.3
|
||||
/-- #### Exercise 2.3
|
||||
|
||||
Show that every member of a set `A` is a subset of `U A`. (This was stated as an
|
||||
example in this section.)
|
||||
|
|
@ -84,7 +84,7 @@ theorem exercise_2_3 {A : Set (Set α)}
|
|||
rw [Set.mem_setOf_eq]
|
||||
exact ⟨x, ⟨hx, hy⟩⟩
|
||||
|
||||
/-- ### Exercise 2.4
|
||||
/-- #### Exercise 2.4
|
||||
|
||||
Show that if `A ⊆ B`, then `⋃ A ⊆ ⋃ B`.
|
||||
-/
|
||||
|
|
@ -96,7 +96,7 @@ theorem exercise_2_4 {A B : Set (Set α)} (h : A ⊆ B) : ⋃₀ A ⊆ ⋃₀ B
|
|||
rw [Set.mem_setOf_eq]
|
||||
exact ⟨t, ⟨h ht, hxt⟩⟩
|
||||
|
||||
/-- ### Exercise 2.5
|
||||
/-- #### Exercise 2.5
|
||||
|
||||
Assume that every member of `𝓐` is a subset of `B`. Show that `⋃ 𝓐 ⊆ B`.
|
||||
-/
|
||||
|
|
@ -108,7 +108,7 @@ theorem exercise_2_5 {𝓐 : Set (Set α)} (h : ∀ x ∈ 𝓐, x ⊆ B)
|
|||
have ⟨t, ⟨ht𝓐, hyt⟩⟩ := hy
|
||||
exact (h t ht𝓐) hyt
|
||||
|
||||
/-- ### Exercise 2.6a
|
||||
/-- #### Exercise 2.6a
|
||||
|
||||
Show that for any set `A`, `⋃ 𝓟 A = A`.
|
||||
-/
|
||||
|
|
@ -125,7 +125,7 @@ theorem exercise_2_6a : ⋃₀ (𝒫 A) = A := by
|
|||
rw [Set.mem_setOf_eq]
|
||||
exact ⟨A, ⟨by rw [Set.mem_setOf_eq], hx⟩⟩
|
||||
|
||||
/-- ### Exercise 2.6b
|
||||
/-- #### Exercise 2.6b
|
||||
|
||||
Show that `A ⊆ 𝓟 ⋃ A`. Under what conditions does equality hold?
|
||||
-/
|
||||
|
|
@ -144,7 +144,7 @@ theorem exercise_2_6b
|
|||
conv => rhs; rw [hB, exercise_2_6a]
|
||||
exact hB
|
||||
|
||||
/-- ### Exercise 2.7a
|
||||
/-- #### Exercise 2.7a
|
||||
|
||||
Show that for any sets `A` and `B`, `𝓟 A ∩ 𝓟 B = 𝓟 (A ∩ B)`.
|
||||
-/
|
||||
|
|
@ -162,9 +162,7 @@ theorem exercise_2_7A
|
|||
intro x hA _
|
||||
exact hA
|
||||
|
||||
-- theorem false_of_false_iff_true : (false ↔ true) → false := by simp
|
||||
|
||||
/-- ### Exercise 2.7b (i)
|
||||
/-- #### Exercise 2.7b (i)
|
||||
|
||||
Show that `𝓟 A ∪ 𝓟 B ⊆ 𝓟 (A ∪ B)`.
|
||||
-/
|
||||
|
|
@ -181,7 +179,7 @@ theorem exercise_2_7b_i
|
|||
rw [Set.mem_setOf_eq]
|
||||
exact Set.subset_union_of_subset_right hB A
|
||||
|
||||
/-- ### Exercise 2.7b (ii)
|
||||
/-- #### Exercise 2.7b (ii)
|
||||
|
||||
Under what conditions does `𝓟 A ∪ 𝓟 B = 𝓟 (A ∪ B)`.?
|
||||
-/
|
||||
|
|
@ -233,7 +231,7 @@ theorem exercise_2_7b_ii
|
|||
refine Or.inl (Set.Subset.trans hx ?_)
|
||||
exact subset_of_eq (Set.right_subset_union_eq_self hB)
|
||||
|
||||
/-- ### Exercise 2.9
|
||||
/-- #### Exercise 2.9
|
||||
|
||||
Give an example of sets `a` and `B` for which `a ∈ B` but `𝓟 a ∉ 𝓟 B`.
|
||||
-/
|
||||
|
|
@ -261,7 +259,7 @@ theorem exercise_2_9 (ha : a = {1}) (hB : B = {{1}})
|
|||
have := h 1
|
||||
simp at this
|
||||
|
||||
/-- ### Exercise 2.10
|
||||
/-- #### Exercise 2.10
|
||||
|
||||
Show that if `a ∈ B`, then `𝓟 a ∈ 𝓟 𝓟 ⋃ B`.
|
||||
-/
|
||||
|
|
@ -274,7 +272,7 @@ theorem exercise_2_10 {B : Set (Set α)} (ha : a ∈ B)
|
|||
rw [← hb, Set.mem_setOf_eq]
|
||||
exact h₂
|
||||
|
||||
/-- ### Exercise 2.11 (i)
|
||||
/-- #### Exercise 2.11 (i)
|
||||
|
||||
Show that for any sets `A` and `B`, `A = (A ∩ B) ∪ (A - B)`.
|
||||
-/
|
||||
|
|
@ -291,7 +289,7 @@ theorem exercise_2_11_i {A B : Set α}
|
|||
· intro hx
|
||||
exact ⟨hx, em (B x)⟩
|
||||
|
||||
/-- ### Exercise 2.11 (ii)
|
||||
/-- #### Exercise 2.11 (ii)
|
||||
|
||||
Show that for any sets `A` and `B`, `A ∪ (B - A) = A ∪ B`.
|
||||
-/
|
||||
|
|
@ -385,7 +383,7 @@ lemma left_diff_eq_singleton_one : (A \ B) \ C = {1} := by
|
|||
| inl y => rw [hx] at y; simp at y
|
||||
| inr y => rw [hx] at y; simp at y
|
||||
|
||||
/-- ### Exercise 2.14
|
||||
/-- #### Exercise 2.14
|
||||
|
||||
Show by example that for some sets `A`, `B`, and `C`, the set `A - (B - C)` is
|
||||
different from `(A - B) - C`.
|
||||
|
|
@ -402,7 +400,7 @@ theorem exercise_2_14 : A \ (B \ C) ≠ (A \ B) \ C := by
|
|||
|
||||
end
|
||||
|
||||
/-- ### Exercise 2.16
|
||||
/-- #### Exercise 2.16
|
||||
|
||||
Simplify:
|
||||
`[(A ∪ B ∪ C) ∩ (A ∪ B)] - [(A ∪ (B - C)) ∩ A]`
|
||||
|
|
@ -414,7 +412,7 @@ theorem exercise_2_16 {A B C : Set α}
|
|||
_ = (A ∪ B) \ A := by rw [Set.union_inter_cancel_left]
|
||||
_ = B \ A := by rw [Set.union_diff_left]
|
||||
|
||||
/-! ### Exercise 2.17
|
||||
/-! #### Exercise 2.17
|
||||
|
||||
Show that the following four conditions are equivalent.
|
||||
|
||||
|
|
@ -450,7 +448,7 @@ theorem exercise_2_17_iii {A B : Set α} (h : A ∪ B = B)
|
|||
theorem exercise_2_17_iv {A B : Set α} (h : A ∩ B = A)
|
||||
: A ⊆ B := Set.inter_eq_left_iff_subset.mp h
|
||||
|
||||
/-- ### Exercise 2.19
|
||||
/-- #### Exercise 2.19
|
||||
|
||||
Is `𝒫 (A - B)` always equal to `𝒫 A - 𝒫 B`? Is it ever equal to `𝒫 A - 𝒫 B`?
|
||||
-/
|
||||
|
|
@ -463,7 +461,7 @@ theorem exercise_2_19 {A B : Set α}
|
|||
have := h ∅
|
||||
exact absurd (this.mp he) ne
|
||||
|
||||
/-- ### Exercise 2.20
|
||||
/-- #### Exercise 2.20
|
||||
|
||||
Let `A`, `B`, and `C` be sets such that `A ∪ B = A ∪ C` and `A ∩ B = A ∩ C`.
|
||||
Show that `B = C`.
|
||||
|
|
@ -489,7 +487,7 @@ theorem exercise_2_20 {A B C : Set α}
|
|||
rw [← hu] at this
|
||||
exact Or.elim this (absurd · hA) (by simp)
|
||||
|
||||
/-- ### Exercise 2.21
|
||||
/-- #### Exercise 2.21
|
||||
|
||||
Show that `⋃ (A ∪ B) = (⋃ A) ∪ (⋃ B)`.
|
||||
-/
|
||||
|
|
@ -513,7 +511,7 @@ theorem exercise_2_21 {A B : Set (Set α)}
|
|||
have ⟨t, ht⟩ : ∃ t, t ∈ B ∧ x ∈ t := hB
|
||||
exact ⟨t, ⟨Set.mem_union_right A ht.left, ht.right⟩⟩
|
||||
|
||||
/-- ### Exercise 2.22
|
||||
/-- #### Exercise 2.22
|
||||
|
||||
Show that if `A` and `B` are nonempty sets, then `⋂ (A ∪ B) = ⋂ A ∩ ⋂ B`.
|
||||
-/
|
||||
|
|
@ -542,7 +540,7 @@ theorem exercise_2_22 {A B : Set (Set α)}
|
|||
· intro hB
|
||||
exact (this t).right hB
|
||||
|
||||
/-- ### Exercise 2.24a
|
||||
/-- #### Exercise 2.24a
|
||||
|
||||
Show that is `𝓐` is nonempty, then `𝒫 (⋂ 𝓐) = ⋂ { 𝒫 X | X ∈ 𝓐 }`.
|
||||
-/
|
||||
|
|
@ -561,7 +559,7 @@ theorem exercise_2_24a {𝓐 : Set (Set α)}
|
|||
_ = { x | ∀ t ∈ { 𝒫 X | X ∈ 𝓐 }, x ∈ t} := by simp
|
||||
_ = ⋂₀ { 𝒫 X | X ∈ 𝓐 } := rfl
|
||||
|
||||
/-- ### Exercise 2.24b
|
||||
/-- #### Exercise 2.24b
|
||||
|
||||
Show that
|
||||
```
|
||||
|
|
@ -603,7 +601,7 @@ theorem exercise_2_24b {𝓐 : Set (Set α)}
|
|||
simp only [Set.mem_setOf_eq, exists_exists_and_eq_and, Set.mem_powerset_iff]
|
||||
exact ⟨⋃₀ 𝓐, ⟨hA, hx⟩⟩
|
||||
|
||||
/-- ### Exercise 2.25
|
||||
/-- #### Exercise 2.25
|
||||
|
||||
Is `A ∪ (⋃ 𝓑)` always the same as `⋃ { A ∪ X | X ∈ 𝓑 }`? If not, then under
|
||||
what conditions does equality hold?
|
||||
|
|
|
|||
|
|
@ -9,7 +9,7 @@ Relations and Functions
|
|||
|
||||
namespace Enderton.Set.Chapter_3
|
||||
|
||||
/-- ### Theorem 3B
|
||||
/-- #### Theorem 3B
|
||||
|
||||
If `x ∈ C` and `y ∈ C`, then `⟨x, y⟩ ∈ 𝒫 𝒫 C`.
|
||||
-/
|
||||
|
|
@ -19,7 +19,7 @@ theorem theorem_3b {C : Set α} (hx : x ∈ C) (hy : y ∈ C)
|
|||
have hxys : {x, y} ⊆ C := Set.mem_mem_imp_pair_subset hx hy
|
||||
exact Set.mem_mem_imp_pair_subset hxs hxys
|
||||
|
||||
/-- ### Exercise 3.1
|
||||
/-- #### Exercise 3.1
|
||||
|
||||
Suppose that we attempted to generalize the Kuratowski definitions of ordered
|
||||
pairs to ordered triples by defining
|
||||
|
|
@ -42,7 +42,7 @@ theorem exercise_3_1 {x y z u v w : ℕ}
|
|||
· rw [hy, hv]
|
||||
simp only
|
||||
|
||||
/-- ### Exercise 3.2a
|
||||
/-- #### Exercise 3.2a
|
||||
|
||||
Show that `A × (B ∪ C) = (A × B) ∪ (A × C)`.
|
||||
-/
|
||||
|
|
@ -58,7 +58,7 @@ theorem exercise_3_2a {A : Set α} {B C : Set β}
|
|||
_ = { p | p ∈ Set.prod A B ∨ (p ∈ Set.prod A C) } := rfl
|
||||
_ = (Set.prod A B) ∪ (Set.prod A C) := rfl
|
||||
|
||||
/-- ### Exercise 3.2b
|
||||
/-- #### Exercise 3.2b
|
||||
|
||||
Show that if `A × B = A × C` and `A ≠ ∅`, then `B = C`.
|
||||
-/
|
||||
|
|
@ -87,7 +87,7 @@ theorem exercise_3_2b {A : Set α} {B C : Set β}
|
|||
have ⟨c, hc⟩ := Set.nonempty_iff_ne_empty.mpr (Ne.symm nC)
|
||||
exact (h (a, c)).mpr ⟨ha, hc⟩
|
||||
|
||||
/-- ### Exercise 3.3
|
||||
/-- #### Exercise 3.3
|
||||
|
||||
Show that `A × ⋃ 𝓑 = ⋃ {A × X | X ∈ 𝓑}`.
|
||||
-/
|
||||
|
|
@ -115,7 +115,7 @@ theorem exercise_3_3 {A : Set (Set α)} {𝓑 : Set (Set β)}
|
|||
· intro ⟨b, h₁, h₂, h₃⟩
|
||||
exact ⟨b, h₁, h₂, h₃⟩
|
||||
|
||||
/-- ### Exercise 3.5a
|
||||
/-- #### Exercise 3.5a
|
||||
|
||||
Assume that `A` and `B` are given sets, and show that there exists a set `C`
|
||||
such that for any `y`,
|
||||
|
|
@ -183,7 +183,7 @@ theorem exercise_3_5a {A : Set α} {B : Set β}
|
|||
rw [hab.right]
|
||||
exact ⟨hab.left, hb⟩
|
||||
|
||||
/-- ### Exercise 3.5b
|
||||
/-- #### Exercise 3.5b
|
||||
|
||||
With `A`, `B`, and `C` as above, show that `A × B = ∪ C`.
|
||||
-/
|
||||
|
|
@ -216,7 +216,7 @@ theorem exercise_3_5b {A : Set α} (B : Set β)
|
|||
rw [← ha] at h
|
||||
exact ⟨h, hb⟩
|
||||
|
||||
/-- ### Theorem 3D
|
||||
/-- #### Theorem 3D
|
||||
|
||||
If `⟨x, y⟩ ∈ A`, then `x` and `y` belong to `⋃ ⋃ A`.
|
||||
-/
|
||||
|
|
@ -228,7 +228,7 @@ theorem theorem_3d {A : Set (Set (Set α))} (h : OrderedPair x y ∈ A)
|
|||
have : {x, y} ⊆ ⋃₀ ⋃₀ A := Chapter_2.exercise_2_3 {x, y} hq
|
||||
exact ⟨this (by simp), this (by simp)⟩
|
||||
|
||||
/-- ### Exercise 3.6
|
||||
/-- #### Exercise 3.6
|
||||
|
||||
Show that a set `A` is a relation **iff** `A ⊆ dom A × ran A`.
|
||||
-/
|
||||
|
|
@ -246,7 +246,7 @@ theorem exercise_3_6 {A : Set.Relation α}
|
|||
]
|
||||
exact ⟨⟨b, ht⟩, ⟨a, ht⟩⟩
|
||||
|
||||
/-- ### Exercise 3.7
|
||||
/-- #### Exercise 3.7
|
||||
|
||||
Show that if `R` is a relation, then `fld R = ⋃ ⋃ R`.
|
||||
-/
|
||||
|
|
@ -329,7 +329,7 @@ section
|
|||
|
||||
open Set.Relation
|
||||
|
||||
/-- ### Exercise 3.8 (i)
|
||||
/-- #### Exercise 3.8 (i)
|
||||
|
||||
Show that for any set `𝓐`:
|
||||
```
|
||||
|
|
@ -355,7 +355,7 @@ theorem exercise_3_8_i {A : Set (Set.Relation α)}
|
|||
· intro ⟨t, ht, y, hx⟩
|
||||
exact ⟨y, t, ht, hx⟩
|
||||
|
||||
/-- ### Exercise 3.8 (ii)
|
||||
/-- #### Exercise 3.8 (ii)
|
||||
|
||||
Show that for any set `𝓐`:
|
||||
```
|
||||
|
|
@ -380,7 +380,7 @@ theorem exercise_3_8_ii {A : Set (Set.Relation α)}
|
|||
· intro ⟨y, ⟨hy, ⟨t, ht⟩⟩⟩
|
||||
exact ⟨t, ⟨y, ⟨hy, ht⟩⟩⟩
|
||||
|
||||
/-- ### Exercise 3.9 (i)
|
||||
/-- #### Exercise 3.9 (i)
|
||||
|
||||
Discuss the result of replacing the union operation by the intersection
|
||||
operation in the preceding problem.
|
||||
|
|
@ -406,7 +406,7 @@ theorem exercise_3_9_i {A : Set (Set.Relation α)}
|
|||
intro _ y hy R hR
|
||||
exact ⟨y, hy R hR⟩
|
||||
|
||||
/-- ### Exercise 3.9 (ii)
|
||||
/-- #### Exercise 3.9 (ii)
|
||||
|
||||
Discuss the result of replacing the union operation by the intersection
|
||||
operation in the preceding problem.
|
||||
|
|
@ -432,7 +432,7 @@ theorem exercise_3_9_ii {A : Set (Set.Relation α)}
|
|||
intro _ y hy R hR
|
||||
exact ⟨y, hy R hR⟩
|
||||
|
||||
/-- ### Theorem 3G (i)
|
||||
/-- #### Theorem 3G (i)
|
||||
|
||||
Assume that `F` is a one-to-one function. If `x ∈ dom F`, then `F⁻¹(F(x)) = x`.
|
||||
-/
|
||||
|
|
@ -446,7 +446,7 @@ theorem theorem_3g_i {F : Set.Relation α}
|
|||
unfold isOneToOne at hF
|
||||
exact (single_valued_eq_unique hF.left hy hy₁).symm
|
||||
|
||||
/-- ### Theorem 3G (ii)
|
||||
/-- #### Theorem 3G (ii)
|
||||
|
||||
Assume that `F` is a one-to-one function. If `y ∈ ran F`, then `F(F⁻¹(y)) = y`.
|
||||
-/
|
||||
|
|
@ -460,7 +460,7 @@ theorem theorem_3g_ii {F : Set.Relation α}
|
|||
unfold isOneToOne at hF
|
||||
exact (single_rooted_eq_unique hF.right hx hx₁).symm
|
||||
|
||||
/-- ### Theorem 3H
|
||||
/-- #### Theorem 3H
|
||||
|
||||
Assume that `F` and `G` are functions. Then
|
||||
```
|
||||
|
|
@ -500,7 +500,7 @@ theorem theorem_3h_dom {F G : Set.Relation α}
|
|||
simp only [Set.mem_setOf_eq]
|
||||
exact ⟨a, ha.left.left, hb⟩
|
||||
|
||||
/-- ### Theorem 3J (a)
|
||||
/-- #### Theorem 3J (a)
|
||||
|
||||
Assume that `F : A → B`, and that `A` is nonempty. There exists a function
|
||||
`G : B → A` (a "left inverse") such that `G ∘ F` is the identity function on `A`
|
||||
|
|
@ -512,7 +512,7 @@ theorem theorem_3j_a {F : Set.Relation α} {A B : Set α}
|
|||
G.mapsInto B A ∧ (∀ p ∈ G.comp F, p.1 = p.2)) ↔ F.isOneToOne := by
|
||||
sorry
|
||||
|
||||
/-- ### Theorem 3J (b)
|
||||
/-- #### Theorem 3J (b)
|
||||
|
||||
Assume that `F : A → B`, and that `A` is nonempty. There exists a function
|
||||
`H : B → A` (a "right inverse") such that `F ∘ H` is the identity function on
|
||||
|
|
|
|||
Loading…
Reference in New Issue