Pointwise operations of sets

This file defines pointwise algebraic operations on sets.

Main declarations

For sets s and t and scalar a:

• s * t: Multiplication, set of all x * y where x ∈ s and y ∈ t.
• s + t: Addition, set of all x + y where x ∈ s and y ∈ t.
• s⁻¹: Inversion, set of all x⁻¹ where x ∈ s.
• -s: Negation, set of all -x where x ∈ s.
• s / t: Division, set of all x / y where x ∈ s and y ∈ t.
• s - t: Subtraction, set of all x - y where x ∈ s and y ∈ t.

For α a semigroup/monoid, Set α is a semigroup/monoid. As an unfortunate side effect, this means that n • s, where n : ℕ, is ambiguous between pointwise scaling and repeated pointwise addition; the former has (2 : ℕ) • {1, 2} = {2, 4}, while the latter has (2 : ℕ) • {1, 2} = {2, 3, 4}. See note [pointwise nat action].

Appropriate definitions and results are also transported to the additive theory via to_additive.

Implementation notes

• The following expressions are considered in simp-normal form in a group: (fun h ↦ h * g) ⁻¹' s, (fun h ↦ g * h) ⁻¹' s, (fun h ↦ h * g⁻¹) ⁻¹' s, (fun h ↦ g⁻¹ * h) ⁻¹' s, s * t, s⁻¹, (1 : Set _) (and similarly for additive variants). Expressions equal to one of these will be simplified.
• We put all instances in the locale Pointwise, so that these instances are not available by default. Note that we do not mark them as reducible (as argued by note [reducible non-instances]) since we expect the locale to be open whenever the instances are actually used (and making the instances reducible changes the behavior of simp.

Tags

set multiplication, set addition, pointwise addition, pointwise multiplication, pointwise subtraction

0/1 as sets

noncomputable def Set.one {α : Type u_2} [One α] : One (Set α)

The set 1 : Set α is defined as {1} in locale Pointwise.

Equations

• Set.one = { one := {1} }

noncomputable def Set.zero {α : Type u_2} [Zero α] : Zero (Set α)

The set 0 : Set α is defined as {0} in locale Pointwise.

Equations

• Set.zero = { zero := {0} }

theorem Set.singleton_one {α : Type u_2} [One α] : {1} = 1

theorem Set.singleton_zero {α : Type u_2} [Zero α] : {0} = 0

@[simp]

theorem Set.mem_one {α : Type u_2} [One α] {a : α} : a ∈ 1 ↔ a = 1

@[simp]

theorem Set.mem_zero {α : Type u_2} [Zero α] {a : α} : a ∈ 0 ↔ a = 0

theorem Set.one_mem_one {α : Type u_2} [One α] : 1 ∈ 1

theorem Set.zero_mem_zero {α : Type u_2} [Zero α] : 0 ∈ 0

@[simp]

theorem Set.one_subset {α : Type u_2} [One α] {s : Set α} : 1 ⊆ s ↔ 1 ∈ s

@[simp]

theorem Set.zero_subset {α : Type u_2} [Zero α] {s : Set α} : 0 ⊆ s ↔ 0 ∈ s

@[simp]

theorem Set.one_nonempty {α : Type u_2} [One α] : Set.Nonempty 1

@[simp]

theorem Set.zero_nonempty {α : Type u_2} [Zero α] : Set.Nonempty 0

@[simp]

theorem Set.image_one {α : Type u_2} {β : Type u_3} [One α] {f : α → β} : f '' 1 = {f 1}

@[simp]

theorem Set.image_zero {α : Type u_2} {β : Type u_3} [Zero α] {f : α → β} : f '' 0 = {f 0}

theorem Set.subset_one_iff_eq {α : Type u_2} [One α] {s : Set α} : s ⊆ 1 ↔ s = ∅ ∨ s = 1

theorem Set.subset_zero_iff_eq {α : Type u_2} [Zero α] {s : Set α} : s ⊆ 0 ↔ s = ∅ ∨ s = 0

theorem Set.Nonempty.subset_one_iff {α : Type u_2} [One α] {s : Set α} (h : s.Nonempty) : s ⊆ 1 ↔ s = 1

theorem Set.Nonempty.subset_zero_iff {α : Type u_2} [Zero α] {s : Set α} (h : s.Nonempty) : s ⊆ 0 ↔ s = 0

noncomputable def Set.singletonOneHom {α : Type u_2} [One α] : OneHom α (Set α)

The singleton operation as a OneHom.

Equations

• Set.singletonOneHom = { toFun := singleton, map_one' := ⋯ }

noncomputable def Set.singletonZeroHom {α : Type u_2} [Zero α] : ZeroHom α (Set α)

The singleton operation as a ZeroHom.

Equations

• Set.singletonZeroHom = { toFun := singleton, map_zero' := ⋯ }

@[simp]

theorem Set.coe_singletonOneHom {α : Type u_2} [One α] : ⇑singletonOneHom = singleton

@[simp]

theorem Set.coe_singletonZeroHom {α : Type u_2} [Zero α] : ⇑singletonZeroHom = singleton

theorem Set.image_op_one {α : Type u_2} [One α] : MulOpposite.op '' 1 = 1

theorem Set.image_op_zero {α : Type u_2} [Zero α] : AddOpposite.op '' 0 = 0

@[simp]

theorem Set.one_prod_one {α : Type u_2} {β : Type u_3} [One α] [One β] : 1 ×ˢ 1 = 1

@[simp]

theorem Set.zero_prod_zero {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] : 0 ×ˢ 0 = 0

@[deprecated Set.zero_prod_zero (since := "2025-03-11")]

theorem Set.zero_sum_zero {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] : 0 ×ˢ 0 = 0

Alias of Set.zero_prod_zero.

Set negation/inversion

def Set.inv {α : Type u_2} [Inv α] : Inv (Set α)

The pointwise inversion of set s⁻¹ is defined as {x | x⁻¹ ∈ s} in locale Pointwise. It is equal to {x⁻¹ | x ∈ s}, see Set.image_inv_eq_inv.

Equations

• Set.inv = { inv := Set.preimage Inv.inv }

def Set.neg {α : Type u_2} [Neg α] : Neg (Set α)

The pointwise negation of set -s is defined as {x | -x ∈ s} in locale Pointwise. It is equal to {-x | x ∈ s}, see Set.image_neg_eq_neg.

Equations

• Set.neg = { neg := Set.preimage Neg.neg }

@[simp]

theorem Set.mem_inv {α : Type u_2} [Inv α] {s : Set α} {a : α} : a ∈ s⁻¹ ↔ a⁻¹ ∈ s

@[simp]

theorem Set.mem_neg {α : Type u_2} [Neg α] {s : Set α} {a : α} : a ∈ -s ↔ -a ∈ s

@[simp]

theorem Set.inv_preimage {α : Type u_2} [Inv α] {s : Set α} : Inv.inv ⁻¹' s = s⁻¹

@[simp]

theorem Set.neg_preimage {α : Type u_2} [Neg α] {s : Set α} : Neg.neg ⁻¹' s = -s

@[simp]

theorem Set.inv_empty {α : Type u_2} [Inv α] : ∅⁻¹ = ∅

@[simp]

theorem Set.neg_empty {α : Type u_2} [Neg α] : -∅ = ∅

@[simp]

theorem Set.inv_univ {α : Type u_2} [Inv α] : univ⁻¹ = univ

@[simp]

theorem Set.neg_univ {α : Type u_2} [Neg α] : -univ = univ

@[simp]

theorem Set.inter_inv {α : Type u_2} [Inv α] {s t : Set α} : (s ∩ t)⁻¹ = s⁻¹ ∩ t⁻¹

@[simp]

theorem Set.inter_neg {α : Type u_2} [Neg α] {s t : Set α} : -(s ∩ t) = -s ∩ -t

@[simp]

theorem Set.union_inv {α : Type u_2} [Inv α] {s t : Set α} : (s ∪ t)⁻¹ = s⁻¹ ∪ t⁻¹

@[simp]

theorem Set.union_neg {α : Type u_2} [Neg α] {s t : Set α} : -(s ∪ t) = -s ∪ -t

@[simp]

theorem Set.compl_inv {α : Type u_2} [Inv α] {s : Set α} : sᶜ⁻¹ = s⁻¹ᶜ

@[simp]

theorem Set.compl_neg {α : Type u_2} [Neg α] {s : Set α} : -sᶜ = (-s)ᶜ

@[simp]

theorem Set.inv_prod {α : Type u_2} {β : Type u_3} [Inv α] [Inv β] (s : Set α) (t : Set β) : (s ×ˢ t)⁻¹ = s⁻¹ ×ˢ t⁻¹

@[simp]

theorem Set.neg_prod {α : Type u_2} {β : Type u_3} [Neg α] [Neg β] (s : Set α) (t : Set β) : -s ×ˢ t = (-s) ×ˢ (-t)

@[deprecated Set.neg_prod (since := "2025-03-11")]

theorem Set.neg_sum {α : Type u_2} {β : Type u_3} [Neg α] [Neg β] (s : Set α) (t : Set β) : -s ×ˢ t = (-s) ×ˢ (-t)

Alias of Set.neg_prod.

theorem Set.inv_mem_inv {α : Type u_2} [InvolutiveInv α] {s : Set α} {a : α} : a⁻¹ ∈ s⁻¹ ↔ a ∈ s

theorem Set.neg_mem_neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} {a : α} : -a ∈ -s ↔ a ∈ s

@[simp]

theorem Set.nonempty_inv {α : Type u_2} [InvolutiveInv α] {s : Set α} : s⁻¹.Nonempty ↔ s.Nonempty

@[simp]

theorem Set.nonempty_neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} : (-s).Nonempty ↔ s.Nonempty

theorem Set.Nonempty.inv {α : Type u_2} [InvolutiveInv α] {s : Set α} (h : s.Nonempty) : s⁻¹.Nonempty

theorem Set.Nonempty.neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} (h : s.Nonempty) : (-s).Nonempty

@[simp]

theorem Set.image_inv_eq_inv {α : Type u_2} [InvolutiveInv α] {s : Set α} : (fun (x : α) => x⁻¹) '' s = s⁻¹

@[simp]

theorem Set.image_neg_eq_neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} : (fun (x : α) => -x) '' s = -s

@[simp]

theorem Set.inv_eq_empty {α : Type u_2} [InvolutiveInv α] {s : Set α} : s⁻¹ = ∅ ↔ s = ∅

@[simp]

theorem Set.neg_eq_empty {α : Type u_2} [InvolutiveNeg α] {s : Set α} : -s = ∅ ↔ s = ∅

noncomputable instance Set.involutiveInv {α : Type u_2} [InvolutiveInv α] : InvolutiveInv (Set α)

Equations

• Set.involutiveInv = { toInv := Set.inv, inv_inv := ⋯ }

noncomputable instance Set.involutiveNeg {α : Type u_2} [InvolutiveNeg α] : InvolutiveNeg (Set α)

Equations

• Set.involutiveNeg = { toNeg := Set.neg, neg_neg := ⋯ }

@[simp]

theorem Set.inv_subset_inv {α : Type u_2} [InvolutiveInv α] {s t : Set α} : s⁻¹ ⊆ t⁻¹ ↔ s ⊆ t

@[simp]

theorem Set.neg_subset_neg {α : Type u_2} [InvolutiveNeg α] {s t : Set α} : -s ⊆ -t ↔ s ⊆ t

theorem Set.inv_subset {α : Type u_2} [InvolutiveInv α] {s t : Set α} : s⁻¹ ⊆ t ↔ s ⊆ t⁻¹

theorem Set.neg_subset {α : Type u_2} [InvolutiveNeg α] {s t : Set α} : -s ⊆ t ↔ s ⊆ -t

@[simp]

theorem Set.inv_singleton {α : Type u_2} [InvolutiveInv α] (a : α) : {a}⁻¹ = {a⁻¹}

@[simp]

theorem Set.neg_singleton {α : Type u_2} [InvolutiveNeg α] (a : α) : -{a} = {-a}

@[simp]

theorem Set.inv_insert {α : Type u_2} [InvolutiveInv α] (a : α) (s : Set α) : (insert a s)⁻¹ = insert a⁻¹ s⁻¹

@[simp]

theorem Set.neg_insert {α : Type u_2} [InvolutiveNeg α] (a : α) (s : Set α) : -insert a s = insert (-a) (-s)

theorem Set.inv_range {α : Type u_2} [InvolutiveInv α] {ι : Sort u_5} {f : ι → α} : (range f)⁻¹ = range fun (i : ι) => (f i)⁻¹

theorem Set.neg_range {α : Type u_2} [InvolutiveNeg α] {ι : Sort u_5} {f : ι → α} : -range f = range fun (i : ι) => -f i

theorem Set.image_op_inv {α : Type u_2} [InvolutiveInv α] {s : Set α} : MulOpposite.op '' s⁻¹ = (MulOpposite.op '' s)⁻¹

theorem Set.image_op_neg {α : Type u_2} [InvolutiveNeg α] {s : Set α} : AddOpposite.op '' (-s) = -AddOpposite.op '' s

Set addition/multiplication

def Set.mul {α : Type u_2} [Mul α] : Mul (Set α)

The pointwise multiplication of sets s * t and t is defined as {x * y | x ∈ s, y ∈ t} in locale Pointwise.

Equations

• Set.mul = { mul := Set.image2 fun (x1 x2 : α) => x1 * x2 }

def Set.add {α : Type u_2} [Add α] : Add (Set α)

The pointwise addition of sets s + t is defined as {x + y | x ∈ s, y ∈ t} in locale Pointwise.

Equations

• Set.add = { add := Set.image2 fun (x1 x2 : α) => x1 + x2 }

@[simp]

theorem Set.image2_mul {α : Type u_2} [Mul α] {s t : Set α} : image2 (fun (x1 x2 : α) => x1 * x2) s t = s * t

@[simp]

theorem Set.image2_add {α : Type u_2} [Add α] {s t : Set α} : image2 (fun (x1 x2 : α) => x1 + x2) s t = s + t

theorem Set.mem_mul {α : Type u_2} [Mul α] {s t : Set α} {a : α} : a ∈ s * t ↔ ∃ x ∈ s, ∃ y ∈ t, x * y = a

theorem Set.mem_add {α : Type u_2} [Add α] {s t : Set α} {a : α} : a ∈ s + t ↔ ∃ x ∈ s, ∃ y ∈ t, x + y = a

theorem Set.mul_mem_mul {α : Type u_2} [Mul α] {s t : Set α} {a b : α} : a ∈ s → b ∈ t → a * b ∈ s * t

theorem Set.add_mem_add {α : Type u_2} [Add α] {s t : Set α} {a b : α} : a ∈ s → b ∈ t → a + b ∈ s + t

theorem Set.image_mul_prod {α : Type u_2} [Mul α] {s t : Set α} : (fun (x : α × α) => x.1 * x.2) '' s ×ˢ t = s * t

theorem Set.add_image_prod {α : Type u_2} [Add α] {s t : Set α} : (fun (x : α × α) => x.1 + x.2) '' s ×ˢ t = s + t

@[simp]

theorem Set.empty_mul {α : Type u_2} [Mul α] {s : Set α} : ∅ * s = ∅

@[simp]

theorem Set.empty_add {α : Type u_2} [Add α] {s : Set α} : ∅ + s = ∅

@[simp]

theorem Set.mul_empty {α : Type u_2} [Mul α] {s : Set α} : s * ∅ = ∅

@[simp]

theorem Set.add_empty {α : Type u_2} [Add α] {s : Set α} : s + ∅ = ∅

@[simp]

theorem Set.mul_eq_empty {α : Type u_2} [Mul α] {s t : Set α} : s * t = ∅ ↔ s = ∅ ∨ t = ∅

@[simp]

theorem Set.add_eq_empty {α : Type u_2} [Add α] {s t : Set α} : s + t = ∅ ↔ s = ∅ ∨ t = ∅

@[simp]

theorem Set.mul_nonempty {α : Type u_2} [Mul α] {s t : Set α} : (s * t).Nonempty ↔ s.Nonempty ∧ t.Nonempty

@[simp]

theorem Set.add_nonempty {α : Type u_2} [Add α] {s t : Set α} : (s + t).Nonempty ↔ s.Nonempty ∧ t.Nonempty

theorem Set.Nonempty.mul {α : Type u_2} [Mul α] {s t : Set α} : s.Nonempty → t.Nonempty → (s * t).Nonempty

theorem Set.Nonempty.add {α : Type u_2} [Add α] {s t : Set α} : s.Nonempty → t.Nonempty → (s + t).Nonempty

theorem Set.Nonempty.of_mul_left {α : Type u_2} [Mul α] {s t : Set α} : (s * t).Nonempty → s.Nonempty

theorem Set.Nonempty.of_add_left {α : Type u_2} [Add α] {s t : Set α} : (s + t).Nonempty → s.Nonempty

theorem Set.Nonempty.of_mul_right {α : Type u_2} [Mul α] {s t : Set α} : (s * t).Nonempty → t.Nonempty

theorem Set.Nonempty.of_add_right {α : Type u_2} [Add α] {s t : Set α} : (s + t).Nonempty → t.Nonempty

@[simp]

theorem Set.mul_singleton {α : Type u_2} [Mul α] {s : Set α} {b : α} : s * {b} = (fun (x : α) => x * b) '' s

@[simp]

theorem Set.add_singleton {α : Type u_2} [Add α] {s : Set α} {b : α} : s + {b} = (fun (x : α) => x + b) '' s

@[simp]

theorem Set.singleton_mul {α : Type u_2} [Mul α] {t : Set α} {a : α} : {a} * t = (fun (x : α) => a * x) '' t

@[simp]

theorem Set.singleton_add {α : Type u_2} [Add α] {t : Set α} {a : α} : {a} + t = (fun (x : α) => a + x) '' t

theorem Set.singleton_mul_singleton {α : Type u_2} [Mul α] {a b : α} : {a} * {b} = {a * b}

theorem Set.singleton_add_singleton {α : Type u_2} [Add α] {a b : α} : {a} + {b} = {a + b}

theorem Set.mul_subset_mul {α : Type u_2} [Mul α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ * s₂ ⊆ t₁ * t₂

theorem Set.add_subset_add {α : Type u_2} [Add α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ + s₂ ⊆ t₁ + t₂

theorem Set.mul_subset_mul_left {α : Type u_2} [Mul α] {s t₁ t₂ : Set α} : t₁ ⊆ t₂ → s * t₁ ⊆ s * t₂

theorem Set.add_subset_add_left {α : Type u_2} [Add α] {s t₁ t₂ : Set α} : t₁ ⊆ t₂ → s + t₁ ⊆ s + t₂

theorem Set.mul_subset_mul_right {α : Type u_2} [Mul α] {s₁ s₂ t : Set α} : s₁ ⊆ s₂ → s₁ * t ⊆ s₂ * t

theorem Set.add_subset_add_right {α : Type u_2} [Add α] {s₁ s₂ t : Set α} : s₁ ⊆ s₂ → s₁ + t ⊆ s₂ + t

instance Set.instMulLeftMono {α : Type u_2} [Mul α] : MulLeftMono (Set α)

instance Set.instAddLeftMono {α : Type u_2} [Add α] : AddLeftMono (Set α)

instance Set.instMulRightMono {α : Type u_2} [Mul α] : MulRightMono (Set α)

instance Set.instAddRightMono {α : Type u_2} [Add α] : AddRightMono (Set α)

theorem Set.mul_subset_iff {α : Type u_2} [Mul α] {s t u : Set α} : s * t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x * y ∈ u

theorem Set.add_subset_iff {α : Type u_2} [Add α] {s t u : Set α} : s + t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x + y ∈ u

theorem Set.union_mul {α : Type u_2} [Mul α] {s₁ s₂ t : Set α} : (s₁ ∪ s₂) * t = s₁ * t ∪ s₂ * t

theorem Set.union_add {α : Type u_2} [Add α] {s₁ s₂ t : Set α} : s₁ ∪ s₂ + t = s₁ + t ∪ (s₂ + t)

theorem Set.mul_union {α : Type u_2} [Mul α] {s t₁ t₂ : Set α} : s * (t₁ ∪ t₂) = s * t₁ ∪ s * t₂

theorem Set.add_union {α : Type u_2} [Add α] {s t₁ t₂ : Set α} : s + (t₁ ∪ t₂) = s + t₁ ∪ (s + t₂)

theorem Set.inter_mul_subset {α : Type u_2} [Mul α] {s₁ s₂ t : Set α} : s₁ ∩ s₂ * t ⊆ s₁ * t ∩ (s₂ * t)

theorem Set.inter_add_subset {α : Type u_2} [Add α] {s₁ s₂ t : Set α} : s₁ ∩ s₂ + t ⊆ (s₁ + t) ∩ (s₂ + t)

theorem Set.mul_inter_subset {α : Type u_2} [Mul α] {s t₁ t₂ : Set α} : s * (t₁ ∩ t₂) ⊆ s * t₁ ∩ (s * t₂)

theorem Set.add_inter_subset {α : Type u_2} [Add α] {s t₁ t₂ : Set α} : s + t₁ ∩ t₂ ⊆ (s + t₁) ∩ (s + t₂)

theorem Set.inter_mul_union_subset_union {α : Type u_2} [Mul α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∩ s₂ * (t₁ ∪ t₂) ⊆ s₁ * t₁ ∪ s₂ * t₂

theorem Set.inter_add_union_subset_union {α : Type u_2} [Add α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∩ s₂ + (t₁ ∪ t₂) ⊆ s₁ + t₁ ∪ (s₂ + t₂)

theorem Set.union_mul_inter_subset_union {α : Type u_2} [Mul α] {s₁ s₂ t₁ t₂ : Set α} : (s₁ ∪ s₂) * (t₁ ∩ t₂) ⊆ s₁ * t₁ ∪ s₂ * t₂

theorem Set.union_add_inter_subset_union {α : Type u_2} [Add α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∪ s₂ + t₁ ∩ t₂ ⊆ s₁ + t₁ ∪ (s₂ + t₂)

noncomputable def Set.singletonMulHom {α : Type u_2} [Mul α] : α →ₙ* Set α

The singleton operation as a MulHom.

Equations

• Set.singletonMulHom = { toFun := singleton, map_mul' := ⋯ }

noncomputable def Set.singletonAddHom {α : Type u_2} [Add α] : α →ₙ+ Set α

The singleton operation as an AddHom.

Equations

• Set.singletonAddHom = { toFun := singleton, map_add' := ⋯ }

@[simp]

theorem Set.coe_singletonMulHom {α : Type u_2} [Mul α] : ⇑singletonMulHom = singleton

@[simp]

theorem Set.coe_singletonAddHom {α : Type u_2} [Add α] : ⇑singletonAddHom = singleton

@[simp]

theorem Set.singletonMulHom_apply {α : Type u_2} [Mul α] (a : α) : singletonMulHom a = {a}

@[simp]

theorem Set.singletonAddHom_apply {α : Type u_2} [Add α] (a : α) : singletonAddHom a = {a}

@[simp]

theorem Set.image_op_mul {α : Type u_2} [Mul α] {s t : Set α} : MulOpposite.op '' (s * t) = MulOpposite.op '' t * MulOpposite.op '' s

@[simp]

theorem Set.image_op_add {α : Type u_2} [Add α] {s t : Set α} : AddOpposite.op '' (s + t) = AddOpposite.op '' t + AddOpposite.op '' s

@[simp]

theorem Set.prod_mul_prod_comm {α : Type u_2} {β : Type u_3} [Mul α] [Mul β] (s₁ s₂ : Set α) (t₁ t₂ : Set β) : s₁ ×ˢ t₁ * s₂ ×ˢ t₂ = (s₁ * s₂) ×ˢ (t₁ * t₂)

@[simp]

theorem Set.prod_add_prod_comm {α : Type u_2} {β : Type u_3} [Add α] [Add β] (s₁ s₂ : Set α) (t₁ t₂ : Set β) : s₁ ×ˢ t₁ + s₂ ×ˢ t₂ = (s₁ + s₂) ×ˢ (t₁ + t₂)

@[deprecated Set.prod_add_prod_comm (since := "2025-03-11")]

theorem Set.sum_add_sum_comm {α : Type u_2} {β : Type u_3} [Add α] [Add β] (s₁ s₂ : Set α) (t₁ t₂ : Set β) : s₁ ×ˢ t₁ + s₂ ×ˢ t₂ = (s₁ + s₂) ×ˢ (t₁ + t₂)

Alias of Set.prod_add_prod_comm.

Set subtraction/division

def Set.div {α : Type u_2} [Div α] : Div (Set α)

The pointwise division of sets s / t is defined as {x / y | x ∈ s, y ∈ t} in locale Pointwise.

Equations

• Set.div = { div := Set.image2 fun (x1 x2 : α) => x1 / x2 }

def Set.sub {α : Type u_2} [Sub α] : Sub (Set α)

The pointwise subtraction of sets s - t is defined as {x - y | x ∈ s, y ∈ t} in locale Pointwise.

Equations

• Set.sub = { sub := Set.image2 fun (x1 x2 : α) => x1 - x2 }

@[simp]

theorem Set.image2_div {α : Type u_2} [Div α] {s t : Set α} : image2 (fun (x1 x2 : α) => x1 / x2) s t = s / t

@[simp]

theorem Set.image2_sub {α : Type u_2} [Sub α] {s t : Set α} : image2 (fun (x1 x2 : α) => x1 - x2) s t = s - t

theorem Set.mem_div {α : Type u_2} [Div α] {s t : Set α} {a : α} : a ∈ s / t ↔ ∃ x ∈ s, ∃ y ∈ t, x / y = a

theorem Set.mem_sub {α : Type u_2} [Sub α] {s t : Set α} {a : α} : a ∈ s - t ↔ ∃ x ∈ s, ∃ y ∈ t, x - y = a

theorem Set.div_mem_div {α : Type u_2} [Div α] {s t : Set α} {a b : α} : a ∈ s → b ∈ t → a / b ∈ s / t

theorem Set.sub_mem_sub {α : Type u_2} [Sub α] {s t : Set α} {a b : α} : a ∈ s → b ∈ t → a - b ∈ s - t

theorem Set.image_div_prod {α : Type u_2} [Div α] {s t : Set α} : (fun (x : α × α) => x.1 / x.2) '' s ×ˢ t = s / t

theorem Set.sub_image_prod {α : Type u_2} [Sub α] {s t : Set α} : (fun (x : α × α) => x.1 - x.2) '' s ×ˢ t = s - t

@[simp]

theorem Set.empty_div {α : Type u_2} [Div α] {s : Set α} : ∅ / s = ∅

@[simp]

theorem Set.empty_sub {α : Type u_2} [Sub α] {s : Set α} : ∅ - s = ∅

@[simp]

theorem Set.div_empty {α : Type u_2} [Div α] {s : Set α} : s / ∅ = ∅

@[simp]

theorem Set.sub_empty {α : Type u_2} [Sub α] {s : Set α} : s - ∅ = ∅

@[simp]

theorem Set.div_eq_empty {α : Type u_2} [Div α] {s t : Set α} : s / t = ∅ ↔ s = ∅ ∨ t = ∅

@[simp]

theorem Set.sub_eq_empty {α : Type u_2} [Sub α] {s t : Set α} : s - t = ∅ ↔ s = ∅ ∨ t = ∅

@[simp]

theorem Set.div_nonempty {α : Type u_2} [Div α] {s t : Set α} : (s / t).Nonempty ↔ s.Nonempty ∧ t.Nonempty

@[simp]

theorem Set.sub_nonempty {α : Type u_2} [Sub α] {s t : Set α} : (s - t).Nonempty ↔ s.Nonempty ∧ t.Nonempty

theorem Set.Nonempty.div {α : Type u_2} [Div α] {s t : Set α} : s.Nonempty → t.Nonempty → (s / t).Nonempty

theorem Set.Nonempty.sub {α : Type u_2} [Sub α] {s t : Set α} : s.Nonempty → t.Nonempty → (s - t).Nonempty

theorem Set.Nonempty.of_div_left {α : Type u_2} [Div α] {s t : Set α} : (s / t).Nonempty → s.Nonempty

theorem Set.Nonempty.of_sub_left {α : Type u_2} [Sub α] {s t : Set α} : (s - t).Nonempty → s.Nonempty

theorem Set.Nonempty.of_div_right {α : Type u_2} [Div α] {s t : Set α} : (s / t).Nonempty → t.Nonempty

theorem Set.Nonempty.of_sub_right {α : Type u_2} [Sub α] {s t : Set α} : (s - t).Nonempty → t.Nonempty

@[simp]

theorem Set.div_singleton {α : Type u_2} [Div α] {s : Set α} {b : α} : s / {b} = (fun (x : α) => x / b) '' s

@[simp]

theorem Set.sub_singleton {α : Type u_2} [Sub α] {s : Set α} {b : α} : s - {b} = (fun (x : α) => x - b) '' s

@[simp]

theorem Set.singleton_div {α : Type u_2} [Div α] {t : Set α} {a : α} : {a} / t = (fun (x1 x2 : α) => x1 / x2) a '' t

@[simp]

theorem Set.singleton_sub {α : Type u_2} [Sub α] {t : Set α} {a : α} : {a} - t = (fun (x1 x2 : α) => x1 - x2) a '' t

theorem Set.singleton_div_singleton {α : Type u_2} [Div α] {a b : α} : {a} / {b} = {a / b}

theorem Set.singleton_sub_singleton {α : Type u_2} [Sub α] {a b : α} : {a} - {b} = {a - b}

theorem Set.div_subset_div {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ / s₂ ⊆ t₁ / t₂

theorem Set.sub_subset_sub {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ⊆ t₁ → s₂ ⊆ t₂ → s₁ - s₂ ⊆ t₁ - t₂

theorem Set.div_subset_div_left {α : Type u_2} [Div α] {s t₁ t₂ : Set α} : t₁ ⊆ t₂ → s / t₁ ⊆ s / t₂

theorem Set.sub_subset_sub_left {α : Type u_2} [Sub α] {s t₁ t₂ : Set α} : t₁ ⊆ t₂ → s - t₁ ⊆ s - t₂

theorem Set.div_subset_div_right {α : Type u_2} [Div α] {s₁ s₂ t : Set α} : s₁ ⊆ s₂ → s₁ / t ⊆ s₂ / t

theorem Set.sub_subset_sub_right {α : Type u_2} [Sub α] {s₁ s₂ t : Set α} : s₁ ⊆ s₂ → s₁ - t ⊆ s₂ - t

theorem Set.div_subset_iff {α : Type u_2} [Div α] {s t u : Set α} : s / t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x / y ∈ u

theorem Set.sub_subset_iff {α : Type u_2} [Sub α] {s t u : Set α} : s - t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x - y ∈ u

theorem Set.union_div {α : Type u_2} [Div α] {s₁ s₂ t : Set α} : (s₁ ∪ s₂) / t = s₁ / t ∪ s₂ / t

theorem Set.union_sub {α : Type u_2} [Sub α] {s₁ s₂ t : Set α} : s₁ ∪ s₂ - t = s₁ - t ∪ (s₂ - t)

theorem Set.div_union {α : Type u_2} [Div α] {s t₁ t₂ : Set α} : s / (t₁ ∪ t₂) = s / t₁ ∪ s / t₂

theorem Set.sub_union {α : Type u_2} [Sub α] {s t₁ t₂ : Set α} : s - (t₁ ∪ t₂) = s - t₁ ∪ (s - t₂)

theorem Set.inter_div_subset {α : Type u_2} [Div α] {s₁ s₂ t : Set α} : s₁ ∩ s₂ / t ⊆ s₁ / t ∩ (s₂ / t)

theorem Set.inter_sub_subset {α : Type u_2} [Sub α] {s₁ s₂ t : Set α} : s₁ ∩ s₂ - t ⊆ (s₁ - t) ∩ (s₂ - t)

theorem Set.div_inter_subset {α : Type u_2} [Div α] {s t₁ t₂ : Set α} : s / (t₁ ∩ t₂) ⊆ s / t₁ ∩ (s / t₂)

theorem Set.sub_inter_subset {α : Type u_2} [Sub α] {s t₁ t₂ : Set α} : s - t₁ ∩ t₂ ⊆ (s - t₁) ∩ (s - t₂)

theorem Set.inter_div_union_subset_union {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∩ s₂ / (t₁ ∪ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂

theorem Set.inter_sub_union_subset_union {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∩ s₂ - (t₁ ∪ t₂) ⊆ s₁ - t₁ ∪ (s₂ - t₂)

theorem Set.union_div_inter_subset_union {α : Type u_2} [Div α] {s₁ s₂ t₁ t₂ : Set α} : (s₁ ∪ s₂) / (t₁ ∩ t₂) ⊆ s₁ / t₁ ∪ s₂ / t₂

theorem Set.union_sub_inter_subset_union {α : Type u_2} [Sub α] {s₁ s₂ t₁ t₂ : Set α} : s₁ ∪ s₂ - t₁ ∩ t₂ ⊆ s₁ - t₁ ∪ (s₂ - t₂)

def Set.NSMul {α : Type u_2} [Zero α] [Add α] : SMul ℕ (Set α)

Repeated pointwise addition (not the same as pointwise repeated addition!) of a Set. See note [pointwise nat action].

Equations

• Set.NSMul = { smul := nsmulRec }

def Set.NPow {α : Type u_2} [One α] [Mul α] : Pow (Set α) ℕ

Repeated pointwise multiplication (not the same as pointwise repeated multiplication!) of a Set. See note [pointwise nat action].

Equations

• Set.NPow = { pow := fun (s : Set α) (n : ℕ) => npowRec n s }

def Set.ZSMul {α : Type u_2} [Zero α] [Add α] [Neg α] : SMul ℤ (Set α)

Repeated pointwise addition/subtraction (not the same as pointwise repeated addition/subtraction!) of a Set. See note [pointwise nat action].

Equations

• Set.ZSMul = { smul := zsmulRec }

def Set.ZPow {α : Type u_2} [One α] [Mul α] [Inv α] : Pow (Set α) ℤ

Repeated pointwise multiplication/division (not the same as pointwise repeated multiplication/division!) of a Set. See note [pointwise nat action].

Equations

• Set.ZPow = { pow := fun (s : Set α) (n : ℤ) => zpowRec npowRec n s }

noncomputable def Set.semigroup {α : Type u_2} [Semigroup α] : Semigroup (Set α)

Set α is a Semigroup under pointwise operations if α is.

Equations

• Set.semigroup = { toMul := Set.mul, mul_assoc := ⋯ }

noncomputable def Set.addSemigroup {α : Type u_2} [AddSemigroup α] : AddSemigroup (Set α)

Set α is an AddSemigroup under pointwise operations if α is.

Equations

• Set.addSemigroup = { toAdd := Set.add, add_assoc := ⋯ }

noncomputable def Set.commSemigroup {α : Type u_2} [CommSemigroup α] : CommSemigroup (Set α)

Set α is a CommSemigroup under pointwise operations if α is.

Equations

• Set.commSemigroup = { toSemigroup := Set.semigroup, mul_comm := ⋯ }

noncomputable def Set.addCommSemigroup {α : Type u_2} [AddCommSemigroup α] : AddCommSemigroup (Set α)

Set α is an AddCommSemigroup under pointwise operations if α is.

Equations

• Set.addCommSemigroup = { toAddSemigroup := Set.addSemigroup, add_comm := ⋯ }

theorem Set.inter_mul_union_subset {α : Type u_2} [CommSemigroup α] {s t : Set α} : s ∩ t * (s ∪ t) ⊆ s * t

theorem Set.inter_add_union_subset {α : Type u_2} [AddCommSemigroup α] {s t : Set α} : s ∩ t + (s ∪ t) ⊆ s + t

theorem Set.union_mul_inter_subset {α : Type u_2} [CommSemigroup α] {s t : Set α} : (s ∪ t) * (s ∩ t) ⊆ s * t

theorem Set.union_add_inter_subset {α : Type u_2} [AddCommSemigroup α] {s t : Set α} : s ∪ t + s ∩ t ⊆ s + t

noncomputable def Set.mulOneClass {α : Type u_2} [MulOneClass α] : MulOneClass (Set α)

Set α is a MulOneClass under pointwise operations if α is.

Equations

• Set.mulOneClass = { toOne := Set.one, toMul := Set.mul, one_mul := ⋯, mul_one := ⋯ }

noncomputable def Set.addZeroClass {α : Type u_2} [AddZeroClass α] : AddZeroClass (Set α)

Set α is an AddZeroClass under pointwise operations if α is.

Equations

• Set.addZeroClass = { toZero := Set.zero, toAdd := Set.add, zero_add := ⋯, add_zero := ⋯ }

theorem Set.subset_mul_left {α : Type u_2} [MulOneClass α] (s : Set α) {t : Set α} (ht : 1 ∈ t) : s ⊆ s * t

theorem Set.subset_add_left {α : Type u_2} [AddZeroClass α] (s : Set α) {t : Set α} (ht : 0 ∈ t) : s ⊆ s + t

theorem Set.subset_mul_right {α : Type u_2} [MulOneClass α] {s : Set α} (t : Set α) (hs : 1 ∈ s) : t ⊆ s * t

theorem Set.subset_add_right {α : Type u_2} [AddZeroClass α] {s : Set α} (t : Set α) (hs : 0 ∈ s) : t ⊆ s + t

noncomputable def Set.singletonMonoidHom {α : Type u_2} [MulOneClass α] : α →* Set α

The singleton operation as a MonoidHom.

Equations

• Set.singletonMonoidHom = { toFun := Set.singletonMulHom.toFun, map_one' := ⋯, map_mul' := ⋯ }

noncomputable def Set.singletonAddMonoidHom {α : Type u_2} [AddZeroClass α] : α →+ Set α

The singleton operation as an AddMonoidHom.

Equations

• Set.singletonAddMonoidHom = { toFun := Set.singletonAddHom.toFun, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Set.coe_singletonMonoidHom {α : Type u_2} [MulOneClass α] : ⇑singletonMonoidHom = singleton

@[simp]

theorem Set.coe_singletonAddMonoidHom {α : Type u_2} [AddZeroClass α] : ⇑singletonAddMonoidHom = singleton

@[simp]

theorem Set.singletonMonoidHom_apply {α : Type u_2} [MulOneClass α] (a : α) : singletonMonoidHom a = {a}

@[simp]

theorem Set.singletonAddMonoidHom_apply {α : Type u_2} [AddZeroClass α] (a : α) : singletonAddMonoidHom a = {a}

noncomputable def Set.monoid {α : Type u_2} [Monoid α] : Monoid (Set α)

Set α is a Monoid under pointwise operations if α is.

Equations

• Set.monoid = { toSemigroup := Set.semigroup, toOne := Set.mulOneClass.toOne, one_mul := ⋯, mul_one := ⋯, npow := npowRecAuto, npow_zero := ⋯, npow_succ := ⋯ }

noncomputable def Set.addMonoid {α : Type u_2} [AddMonoid α] : AddMonoid (Set α)

Set α is an AddMonoid under pointwise operations if α is.

Equations

• Set.addMonoid = { toAddSemigroup := Set.addSemigroup, toZero := Set.addZeroClass.toZero, zero_add := ⋯, add_zero := ⋯, nsmul := nsmulRecAuto, nsmul_zero := ⋯, nsmul_succ := ⋯ }

theorem Set.pow_right_monotone {α : Type u_2} [Monoid α] {s : Set α} (hs : 1 ∈ s) : Monotone fun (x : ℕ) => s ^ x

theorem Set.nsmul_right_monotone {α : Type u_2} [AddMonoid α] {s : Set α} (hs : 0 ∈ s) : Monotone fun (x : ℕ) => x • s

theorem Set.pow_subset_pow_left {α : Type u_2} [Monoid α] {s t : Set α} {n : ℕ} (hst : s ⊆ t) : s ^ n ⊆ t ^ n

theorem Set.nsmul_subset_nsmul_left {α : Type u_2} [AddMonoid α] {s t : Set α} {n : ℕ} (hst : s ⊆ t) : n • s ⊆ n • t

theorem Set.pow_subset_pow_right {α : Type u_2} [Monoid α] {s : Set α} {m n : ℕ} (hs : 1 ∈ s) (hmn : m ≤ n) : s ^ m ⊆ s ^ n

theorem Set.nsmul_subset_nsmul_right {α : Type u_2} [AddMonoid α] {s : Set α} {m n : ℕ} (hs : 0 ∈ s) (hmn : m ≤ n) : m • s ⊆ n • s

theorem Set.pow_subset_pow {α : Type u_2} [Monoid α] {s t : Set α} {m n : ℕ} (hst : s ⊆ t) (ht : 1 ∈ t) (hmn : m ≤ n) : s ^ m ⊆ t ^ n

theorem Set.nsmul_subset_nsmul {α : Type u_2} [AddMonoid α] {s t : Set α} {m n : ℕ} (hst : s ⊆ t) (ht : 0 ∈ t) (hmn : m ≤ n) : m • s ⊆ n • t

theorem Set.subset_pow {α : Type u_2} [Monoid α] {s : Set α} {n : ℕ} (hs : 1 ∈ s) (hn : n ≠ 0) : s ⊆ s ^ n

theorem Set.subset_nsmul {α : Type u_2} [AddMonoid α] {s : Set α} {n : ℕ} (hs : 0 ∈ s) (hn : n ≠ 0) : s ⊆ n • s

@[deprecated Set.pow_subset_pow_right (since := "2024-11-19")]

theorem Set.pow_subset_pow_of_one_mem {α : Type u_2} [Monoid α] {s : Set α} {m n : ℕ} (hs : 1 ∈ s) (hmn : m ≤ n) : s ^ m ⊆ s ^ n

Alias of Set.pow_subset_pow_right.

@[deprecated Set.nsmul_subset_nsmul_right (since := "2024-11-19")]

theorem Set.nsmul_subset_nsmul_of_zero_mem {α : Type u_2} [AddMonoid α] {s : Set α} {m n : ℕ} (hs : 0 ∈ s) (hmn : m ≤ n) : m • s ⊆ n • s

Alias of Set.nsmul_subset_nsmul_right.

theorem Set.pow_subset_pow_mul_of_sq_subset_mul {α : Type u_2} [Monoid α] {s t : Set α} {n : ℕ} (hst : s ^ 2 ⊆ t * s) (hn : n ≠ 0) : s ^ n ⊆ t ^ (n - 1) * s

theorem Set.nsmul_subset_nsmul_add_of_sq_subset_add {α : Type u_2} [AddMonoid α] {s t : Set α} {n : ℕ} (hst : 2 • s ⊆ t + s) (hn : n ≠ 0) : n • s ⊆ (n - 1) • t + s

@[simp]

theorem Set.empty_pow {α : Type u_2} [Monoid α] {n : ℕ} (hn : n ≠ 0) : ∅ ^ n = ∅

@[simp]

theorem Set.nsmul_empty {α : Type u_2} [AddMonoid α] {n : ℕ} (hn : n ≠ 0) : n • ∅ = ∅

theorem Set.Nonempty.pow {α : Type u_2} [Monoid α] {s : Set α} (hs : s.Nonempty) {n : ℕ} : (s ^ n).Nonempty

theorem Set.Nonempty.nsmul {α : Type u_2} [AddMonoid α] {s : Set α} (hs : s.Nonempty) {n : ℕ} : (n • s).Nonempty

@[simp]

theorem Set.pow_eq_empty {α : Type u_2} [Monoid α] {s : Set α} {n : ℕ} : s ^ n = ∅ ↔ s = ∅ ∧ n ≠ 0

@[simp]

theorem Set.nsmul_eq_empty {α : Type u_2} [AddMonoid α] {s : Set α} {n : ℕ} : n • s = ∅ ↔ s = ∅ ∧ n ≠ 0

@[simp]

theorem Set.singleton_pow {α : Type u_2} [Monoid α] (a : α) (n : ℕ) : {a} ^ n = {a ^ n}

@[simp]

theorem Set.nsmul_singleton {α : Type u_2} [AddMonoid α] (a : α) (n : ℕ) : n • {a} = {n • a}

theorem Set.pow_mem_pow {α : Type u_2} [Monoid α] {s : Set α} {a : α} {n : ℕ} (ha : a ∈ s) : a ^ n ∈ s ^ n

theorem Set.nsmul_mem_nsmul {α : Type u_2} [AddMonoid α] {s : Set α} {a : α} {n : ℕ} (ha : a ∈ s) : n • a ∈ n • s

theorem Set.one_mem_pow {α : Type u_2} [Monoid α] {s : Set α} {n : ℕ} (hs : 1 ∈ s) : 1 ∈ s ^ n

theorem Set.zero_mem_nsmul {α : Type u_2} [AddMonoid α] {s : Set α} {n : ℕ} (hs : 0 ∈ s) : 0 ∈ n • s

theorem Set.inter_pow_subset {α : Type u_2} [Monoid α] {s t : Set α} {n : ℕ} : (s ∩ t) ^ n ⊆ s ^ n ∩ t ^ n

theorem Set.inter_nsmul_subset {α : Type u_2} [AddMonoid α] {s t : Set α} {n : ℕ} : n • (s ∩ t) ⊆ n • s ∩ n • t

theorem Set.mul_univ_of_one_mem {α : Type u_2} [Monoid α] {s : Set α} (hs : 1 ∈ s) : s * univ = univ

theorem Set.add_univ_of_zero_mem {α : Type u_2} [AddMonoid α] {s : Set α} (hs : 0 ∈ s) : s + univ = univ

theorem Set.univ_mul_of_one_mem {α : Type u_2} [Monoid α] {t : Set α} (ht : 1 ∈ t) : univ * t = univ

theorem Set.univ_add_of_zero_mem {α : Type u_2} [AddMonoid α] {t : Set α} (ht : 0 ∈ t) : univ + t = univ

@[simp]

theorem Set.univ_mul_univ {α : Type u_2} [Monoid α] : univ * univ = univ

@[simp]

theorem Set.univ_add_univ {α : Type u_2} [AddMonoid α] : univ + univ = univ

@[simp]

theorem Set.univ_pow {α : Type u_2} [Monoid α] {n : ℕ} : n ≠ 0 → univ ^ n = univ

@[simp]

theorem Set.nsmul_univ {α : Type u_2} [AddMonoid α] {n : ℕ} : n ≠ 0 → n • univ = univ

theorem IsUnit.set {α : Type u_2} [Monoid α] {a : α} : IsUnit a → IsUnit {a}

theorem IsAddUnit.set {α : Type u_2} [AddMonoid α] {a : α} : IsAddUnit a → IsAddUnit {a}

theorem Set.prod_pow {α : Type u_2} {β : Type u_3} [Monoid α] [Monoid β] (s : Set α) (t : Set β) (n : ℕ) : s ×ˢ t ^ n = (s ^ n) ×ˢ (t ^ n)

theorem Set.nsmul_prod {α : Type u_2} {β : Type u_3} [AddMonoid α] [AddMonoid β] (s : Set α) (t : Set β) (n : ℕ) : n • s ×ˢ t = (n • s) ×ˢ (n • t)

@[deprecated Set.nsmul_prod (since := "2025-02-17")]

theorem Set.sum_nsmul {α : Type u_2} {β : Type u_3} [AddMonoid α] [AddMonoid β] (s : Set α) (t : Set β) (n : ℕ) : n • s ×ˢ t = (n • s) ×ˢ (n • t)

Alias of Set.nsmul_prod.

theorem Set.Nontrivial.mul_left {α : Type u_2} [Mul α] [IsLeftCancelMul α] {s t : Set α} : t.Nontrivial → s.Nonempty → (s * t).Nontrivial

theorem Set.Nontrivial.add_left {α : Type u_2} [Add α] [IsLeftCancelAdd α] {s t : Set α} : t.Nontrivial → s.Nonempty → (s + t).Nontrivial

theorem Set.Nontrivial.mul {α : Type u_2} [Mul α] [IsLeftCancelMul α] {s t : Set α} (hs : s.Nontrivial) (ht : t.Nontrivial) : (s * t).Nontrivial

theorem Set.Nontrivial.add {α : Type u_2} [Add α] [IsLeftCancelAdd α] {s t : Set α} (hs : s.Nontrivial) (ht : t.Nontrivial) : (s + t).Nontrivial

theorem Set.Nontrivial.mul_right {α : Type u_2} [Mul α] [IsRightCancelMul α] {s t : Set α} : s.Nontrivial → t.Nonempty → (s * t).Nontrivial

theorem Set.Nontrivial.add_right {α : Type u_2} [Add α] [IsRightCancelAdd α] {s t : Set α} : s.Nontrivial → t.Nonempty → (s + t).Nontrivial

theorem Set.Nontrivial.pow {α : Type u_2} [CancelMonoid α] {s : Set α} (hs : s.Nontrivial) {n : ℕ} : n ≠ 0 → (s ^ n).Nontrivial

theorem Set.Nontrivial.nsmul {α : Type u_2} [AddCancelMonoid α] {s : Set α} (hs : s.Nontrivial) {n : ℕ} : n ≠ 0 → (n • s).Nontrivial

noncomputable def Set.commMonoid {α : Type u_2} [CommMonoid α] : CommMonoid (Set α)

Set α is a CommMonoid under pointwise operations if α is.

Equations

• Set.commMonoid = { toMonoid := Set.monoid, mul_comm := ⋯ }

noncomputable def Set.addCommMonoid {α : Type u_2} [AddCommMonoid α] : AddCommMonoid (Set α)

Set α is an AddCommMonoid under pointwise operations if α is.

Equations

• Set.addCommMonoid = { toAddMonoid := Set.addMonoid, add_comm := ⋯ }

theorem Set.mul_eq_one_iff {α : Type u_2} [DivisionMonoid α] {s t : Set α} : s * t = 1 ↔ ∃ (a : α) (b : α), s = {a} ∧ t = {b} ∧ a * b = 1

theorem Set.add_eq_zero_iff {α : Type u_2} [SubtractionMonoid α] {s t : Set α} : s + t = 0 ↔ ∃ (a : α) (b : α), s = {a} ∧ t = {b} ∧ a + b = 0

noncomputable def Set.divisionMonoid {α : Type u_2} [DivisionMonoid α] : DivisionMonoid (Set α)

Set α is a division monoid under pointwise operations if α is.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def Set.subtractionMonoid {α : Type u_2} [SubtractionMonoid α] : SubtractionMonoid (Set α)

Set α is a subtraction monoid under pointwise operations if α is.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Set.isUnit_iff {α : Type u_2} [DivisionMonoid α] {s : Set α} : IsUnit s ↔ ∃ (a : α), s = {a} ∧ IsUnit a

@[simp]

theorem Set.isAddUnit_iff {α : Type u_2} [SubtractionMonoid α] {s : Set α} : IsAddUnit s ↔ ∃ (a : α), s = {a} ∧ IsAddUnit a

@[simp]

theorem Set.univ_div_univ {α : Type u_2} [DivisionMonoid α] : univ / univ = univ

@[simp]

theorem Set.univ_sub_univ {α : Type u_2} [SubtractionMonoid α] : univ - univ = univ

theorem Set.subset_div_left {α : Type u_2} [DivisionMonoid α] {s t : Set α} (ht : 1 ∈ t) : s ⊆ s / t

theorem Set.subset_sub_left {α : Type u_2} [SubtractionMonoid α] {s t : Set α} (ht : 0 ∈ t) : s ⊆ s - t

theorem Set.inv_subset_div_right {α : Type u_2} [DivisionMonoid α] {s t : Set α} (hs : 1 ∈ s) : t⁻¹ ⊆ s / t

theorem Set.neg_subset_sub_right {α : Type u_2} [SubtractionMonoid α] {s t : Set α} (hs : 0 ∈ s) : -t ⊆ s - t

@[simp]

theorem Set.empty_zpow {α : Type u_2} [DivisionMonoid α] {n : ℤ} (hn : n ≠ 0) : ∅ ^ n = ∅

@[simp]

theorem Set.zsmul_empty {α : Type u_2} [SubtractionMonoid α] {n : ℤ} (hn : n ≠ 0) : n • ∅ = ∅

theorem Set.Nonempty.zpow {α : Type u_2} [DivisionMonoid α] {s : Set α} (hs : s.Nonempty) {n : ℤ} : (s ^ n).Nonempty

theorem Set.Nonempty.zsmul {α : Type u_2} [SubtractionMonoid α] {s : Set α} (hs : s.Nonempty) {n : ℤ} : (n • s).Nonempty

@[simp]

theorem Set.zpow_eq_empty {α : Type u_2} [DivisionMonoid α] {s : Set α} {n : ℤ} : s ^ n = ∅ ↔ s = ∅ ∧ n ≠ 0

@[simp]

theorem Set.zsmul_eq_empty {α : Type u_2} [SubtractionMonoid α] {s : Set α} {n : ℤ} : n • s = ∅ ↔ s = ∅ ∧ n ≠ 0

@[simp]

theorem Set.singleton_zpow {α : Type u_2} [DivisionMonoid α] (a : α) (n : ℤ) : {a} ^ n = {a ^ n}

@[simp]

theorem Set.zsmul_singleton {α : Type u_2} [SubtractionMonoid α] (a : α) (n : ℤ) : n • {a} = {n • a}

noncomputable def Set.divisionCommMonoid {α : Type u_2} [DivisionCommMonoid α] : DivisionCommMonoid (Set α)

Set α is a commutative division monoid under pointwise operations if α is.

Equations

• Set.divisionCommMonoid = { toDivisionMonoid := Set.divisionMonoid, mul_comm := ⋯ }

noncomputable def Set.subtractionCommMonoid {α : Type u_2} [SubtractionCommMonoid α] : SubtractionCommMonoid (Set α)

Set α is a commutative subtraction monoid under pointwise operations if α is.

Equations

• Set.subtractionCommMonoid = { toSubtractionMonoid := Set.subtractionMonoid, add_comm := ⋯ }

Note that Set is not a Group because s / s ≠ 1 in general.

@[simp]

theorem Set.one_mem_div_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∈ s / t ↔ ¬Disjoint s t

@[simp]

theorem Set.zero_mem_sub_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∈ s - t ↔ ¬Disjoint s t

@[simp]

theorem Set.one_mem_inv_mul_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∈ t⁻¹ * s ↔ ¬Disjoint s t

@[simp]

theorem Set.zero_mem_neg_add_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∈ -t + s ↔ ¬Disjoint s t

theorem Set.one_notMem_div_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∉ s / t ↔ Disjoint s t

theorem Set.zero_notMem_sub_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∉ s - t ↔ Disjoint s t

@[deprecated Set.zero_notMem_sub_iff (since := "2025-05-23")]

theorem Set.not_zero_mem_sub_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∉ s - t ↔ Disjoint s t

Alias of Set.zero_notMem_sub_iff.

@[deprecated Set.one_notMem_div_iff (since := "2025-05-23")]

theorem Set.not_one_mem_div_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∉ s / t ↔ Disjoint s t

Alias of Set.one_notMem_div_iff.

theorem Set.one_notMem_inv_mul_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∉ t⁻¹ * s ↔ Disjoint s t

theorem Set.zero_notMem_neg_add_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∉ -t + s ↔ Disjoint s t

@[deprecated Set.zero_notMem_neg_add_iff (since := "2025-05-23")]

theorem Set.not_zero_mem_neg_add_iff {α : Type u_2} [AddGroup α] {s t : Set α} : 0 ∉ -t + s ↔ Disjoint s t

Alias of Set.zero_notMem_neg_add_iff.

@[deprecated Set.one_notMem_inv_mul_iff (since := "2025-05-23")]

theorem Set.not_one_mem_inv_mul_iff {α : Type u_2} [Group α] {s t : Set α} : 1 ∉ t⁻¹ * s ↔ Disjoint s t

Alias of Set.one_notMem_inv_mul_iff.

theorem Disjoint.one_notMem_div_set {α : Type u_2} [Group α] {s t : Set α} : Disjoint s t → 1 ∉ s / t

Alias of the reverse direction of Set.one_notMem_div_iff.

theorem Disjoint.zero_notMem_sub_set {α : Type u_2} [AddGroup α] {s t : Set α} : Disjoint s t → 0 ∉ s - t

@[deprecated Disjoint.zero_notMem_sub_set (since := "2025-05-23")]

theorem Disjoint.zero_not_mem_sub_set {α : Type u_2} [AddGroup α] {s t : Set α} : Disjoint s t → 0 ∉ s - t

Alias of Disjoint.zero_notMem_sub_set.

@[deprecated Disjoint.one_notMem_div_set (since := "2025-05-23")]

theorem Disjoint.one_not_mem_div_set {α : Type u_2} [Group α] {s t : Set α} : Disjoint s t → 1 ∉ s / t

Alias of the reverse direction of Set.one_notMem_div_iff.

---

Alias of the reverse direction of Set.one_notMem_div_iff.

theorem Set.Nonempty.one_mem_div {α : Type u_2} [Group α] {s : Set α} (h : s.Nonempty) : 1 ∈ s / s

theorem Set.Nonempty.zero_mem_sub {α : Type u_2} [AddGroup α] {s : Set α} (h : s.Nonempty) : 0 ∈ s - s

theorem Set.isUnit_singleton {α : Type u_2} [Group α] (a : α) : IsUnit {a}

theorem Set.isAddUnit_singleton {α : Type u_2} [AddGroup α] (a : α) : IsAddUnit {a}

@[simp]

theorem Set.isUnit_iff_singleton {α : Type u_2} [Group α] {s : Set α} : IsUnit s ↔ ∃ (a : α), s = {a}

@[simp]

theorem Set.isAddUnit_iff_singleton {α : Type u_2} [AddGroup α] {s : Set α} : IsAddUnit s ↔ ∃ (a : α), s = {a}

@[simp]

theorem Set.image_mul_left {α : Type u_2} [Group α] {t : Set α} {a : α} : (fun (x : α) => a * x) '' t = (fun (x : α) => a⁻¹ * x) ⁻¹' t

@[simp]

theorem Set.image_add_left {α : Type u_2} [AddGroup α] {t : Set α} {a : α} : (fun (x : α) => a + x) '' t = (fun (x : α) => -a + x) ⁻¹' t

@[simp]

theorem Set.image_mul_right {α : Type u_2} [Group α] {t : Set α} {b : α} : (fun (x : α) => x * b) '' t = (fun (x : α) => x * b⁻¹) ⁻¹' t

@[simp]

theorem Set.image_add_right {α : Type u_2} [AddGroup α] {t : Set α} {b : α} : (fun (x : α) => x + b) '' t = (fun (x : α) => x + -b) ⁻¹' t

theorem Set.image_mul_left' {α : Type u_2} [Group α] {t : Set α} {a : α} : (fun (x : α) => a⁻¹ * x) '' t = (fun (x : α) => a * x) ⁻¹' t

theorem Set.image_add_left' {α : Type u_2} [AddGroup α] {t : Set α} {a : α} : (fun (x : α) => -a + x) '' t = (fun (x : α) => a + x) ⁻¹' t

theorem Set.image_mul_right' {α : Type u_2} [Group α] {t : Set α} {b : α} : (fun (x : α) => x * b⁻¹) '' t = (fun (x : α) => x * b) ⁻¹' t

theorem Set.image_add_right' {α : Type u_2} [AddGroup α] {t : Set α} {b : α} : (fun (x : α) => x + -b) '' t = (fun (x : α) => x + b) ⁻¹' t

@[simp]

theorem Set.preimage_mul_left_singleton {α : Type u_2} [Group α] {a b : α} : (fun (x : α) => a * x) ⁻¹' {b} = {a⁻¹ * b}

@[simp]

theorem Set.preimage_add_left_singleton {α : Type u_2} [AddGroup α] {a b : α} : (fun (x : α) => a + x) ⁻¹' {b} = {-a + b}

@[simp]

theorem Set.preimage_mul_right_singleton {α : Type u_2} [Group α] {a b : α} : (fun (x : α) => x * a) ⁻¹' {b} = {b * a⁻¹}

@[simp]

theorem Set.preimage_add_right_singleton {α : Type u_2} [AddGroup α] {a b : α} : (fun (x : α) => x + a) ⁻¹' {b} = {b + -a}

@[simp]

theorem Set.preimage_mul_left_one {α : Type u_2} [Group α] {a : α} : (fun (x : α) => a * x) ⁻¹' 1 = {a⁻¹}

@[simp]

theorem Set.preimage_add_left_zero {α : Type u_2} [AddGroup α] {a : α} : (fun (x : α) => a + x) ⁻¹' 0 = {-a}

@[simp]

theorem Set.preimage_mul_right_one {α : Type u_2} [Group α] {b : α} : (fun (x : α) => x * b) ⁻¹' 1 = {b⁻¹}

@[simp]

theorem Set.preimage_add_right_zero {α : Type u_2} [AddGroup α] {b : α} : (fun (x : α) => x + b) ⁻¹' 0 = {-b}

theorem Set.preimage_mul_left_one' {α : Type u_2} [Group α] {a : α} : (fun (x : α) => a⁻¹ * x) ⁻¹' 1 = {a}

theorem Set.preimage_add_left_zero' {α : Type u_2} [AddGroup α] {a : α} : (fun (x : α) => -a + x) ⁻¹' 0 = {a}

theorem Set.preimage_mul_right_one' {α : Type u_2} [Group α] {b : α} : (fun (x : α) => x * b⁻¹) ⁻¹' 1 = {b}

theorem Set.preimage_add_right_zero' {α : Type u_2} [AddGroup α] {b : α} : (fun (x : α) => x + -b) ⁻¹' 0 = {b}

@[simp]

theorem Set.mul_univ {α : Type u_2} [Group α] {s : Set α} (hs : s.Nonempty) : s * univ = univ

@[simp]

theorem Set.add_univ {α : Type u_2} [AddGroup α] {s : Set α} (hs : s.Nonempty) : s + univ = univ

@[simp]

theorem Set.univ_mul {α : Type u_2} [Group α] {t : Set α} (ht : t.Nonempty) : univ * t = univ

@[simp]

theorem Set.univ_add {α : Type u_2} [AddGroup α] {t : Set α} (ht : t.Nonempty) : univ + t = univ

theorem Set.image_inv {F : Type u_1} {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F) (s : Set α) : ⇑f '' s⁻¹ = (⇑f '' s)⁻¹

theorem Set.image_neg {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) (s : Set α) : ⇑f '' (-s) = -⇑f '' s

theorem Set.image_mul {F : Type u_1} {α : Type u_2} {β : Type u_3} [Mul α] [Mul β] [FunLike F α β] [MulHomClass F α β] (m : F) {s t : Set α} : ⇑m '' (s * t) = ⇑m '' s * ⇑m '' t

theorem Set.image_add {F : Type u_1} {α : Type u_2} {β : Type u_3} [Add α] [Add β] [FunLike F α β] [AddHomClass F α β] (m : F) {s t : Set α} : ⇑m '' (s + t) = ⇑m '' s + ⇑m '' t

theorem Set.mul_subset_range {F : Type u_1} {α : Type u_2} {β : Type u_3} [Mul α] [Mul β] [FunLike F α β] [MulHomClass F α β] (m : F) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : s * t ⊆ range ⇑m

theorem Set.add_subset_range {F : Type u_1} {α : Type u_2} {β : Type u_3} [Add α] [Add β] [FunLike F α β] [AddHomClass F α β] (m : F) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : s + t ⊆ range ⇑m

theorem Set.preimage_mul_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [Mul α] [Mul β] [FunLike F α β] [MulHomClass F α β] (m : F) {s t : Set β} : ⇑m ⁻¹' s * ⇑m ⁻¹' t ⊆ ⇑m ⁻¹' (s * t)

theorem Set.preimage_add_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [Add α] [Add β] [FunLike F α β] [AddHomClass F α β] (m : F) {s t : Set β} : ⇑m ⁻¹' s + ⇑m ⁻¹' t ⊆ ⇑m ⁻¹' (s + t)

theorem Set.preimage_mul {F : Type u_1} {α : Type u_2} {β : Type u_3} [Mul α] [Mul β] [FunLike F α β] [MulHomClass F α β] (m : F) (hm : Function.Injective ⇑m) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : ⇑m ⁻¹' (s * t) = ⇑m ⁻¹' s * ⇑m ⁻¹' t

theorem Set.preimage_add {F : Type u_1} {α : Type u_2} {β : Type u_3} [Add α] [Add β] [FunLike F α β] [AddHomClass F α β] (m : F) (hm : Function.Injective ⇑m) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : ⇑m ⁻¹' (s + t) = ⇑m ⁻¹' s + ⇑m ⁻¹' t

theorem Set.image_pow_of_ne_zero {F : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid α] [Monoid β] [FunLike F α β] [MulHomClass F α β] {n : ℕ} : n ≠ 0 → ∀ (f : F) (s : Set α), ⇑f '' (s ^ n) = (⇑f '' s) ^ n

theorem Set.image_nsmul_of_ne_zero {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddMonoid α] [AddMonoid β] [FunLike F α β] [AddHomClass F α β] {n : ℕ} : n ≠ 0 → ∀ (f : F) (s : Set α), ⇑f '' (n • s) = n • ⇑f '' s

theorem Set.image_pow {F : Type u_1} {α : Type u_2} {β : Type u_3} [Monoid α] [Monoid β] [FunLike F α β] [MonoidHomClass F α β] (f : F) (s : Set α) (n : ℕ) : ⇑f '' (s ^ n) = (⇑f '' s) ^ n

theorem Set.image_nsmul {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddMonoid α] [AddMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (f : F) (s : Set α) (n : ℕ) : ⇑f '' (n • s) = n • ⇑f '' s

theorem Set.image_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] [FunLike F α β] [MonoidHomClass F α β] (m : F) {s t : Set α} : ⇑m '' (s / t) = ⇑m '' s / ⇑m '' t

theorem Set.image_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (m : F) {s t : Set α} : ⇑m '' (s - t) = ⇑m '' s - ⇑m '' t

theorem Set.div_subset_range {F : Type u_1} {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] [FunLike F α β] [MonoidHomClass F α β] (m : F) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : s / t ⊆ range ⇑m

theorem Set.sub_subset_range {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (m : F) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : s - t ⊆ range ⇑m

theorem Set.preimage_div_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] [FunLike F α β] [MonoidHomClass F α β] (m : F) {s t : Set β} : ⇑m ⁻¹' s / ⇑m ⁻¹' t ⊆ ⇑m ⁻¹' (s / t)

theorem Set.preimage_sub_preimage_subset {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (m : F) {s t : Set β} : ⇑m ⁻¹' s - ⇑m ⁻¹' t ⊆ ⇑m ⁻¹' (s - t)

theorem Set.preimage_div {F : Type u_1} {α : Type u_2} {β : Type u_3} [Group α] [DivisionMonoid β] [FunLike F α β] [MonoidHomClass F α β] (m : F) (hm : Function.Injective ⇑m) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : ⇑m ⁻¹' (s / t) = ⇑m ⁻¹' s / ⇑m ⁻¹' t

theorem Set.preimage_sub {F : Type u_1} {α : Type u_2} {β : Type u_3} [AddGroup α] [SubtractionMonoid β] [FunLike F α β] [AddMonoidHomClass F α β] (m : F) (hm : Function.Injective ⇑m) {s t : Set β} (hs : s ⊆ range ⇑m) (ht : t ⊆ range ⇑m) : ⇑m ⁻¹' (s - t) = ⇑m ⁻¹' s - ⇑m ⁻¹' t

@[simp]

theorem Set.inv_pi {ι : Type u_5} {α : ι → Type u_6} [(i : ι) → Inv (α i)] (s : Set ι) (t : (i : ι) → Set (α i)) : (s.pi t)⁻¹ = s.pi fun (i : ι) => (t i)⁻¹

@[simp]

theorem Set.neg_pi {ι : Type u_5} {α : ι → Type u_6} [(i : ι) → Neg (α i)] (s : Set ι) (t : (i : ι) → Set (α i)) : -s.pi t = s.pi fun (i : ι) => -t i

theorem Set.MapsTo.mul {α : Type u_2} {β : Type u_3} [Mul β] {A : Set α} {B₁ B₂ : Set β} {f₁ f₂ : α → β} (h₁ : MapsTo f₁ A B₁) (h₂ : MapsTo f₂ A B₂) : MapsTo (f₁ * f₂) A (B₁ * B₂)

theorem Set.MapsTo.add {α : Type u_2} {β : Type u_3} [Add β] {A : Set α} {B₁ B₂ : Set β} {f₁ f₂ : α → β} (h₁ : MapsTo f₁ A B₁) (h₂ : MapsTo f₂ A B₂) : MapsTo (f₁ + f₂) A (B₁ + B₂)

theorem Set.MapsTo.inv {α : Type u_2} {β : Type u_3} [InvolutiveInv β] {A : Set α} {B : Set β} {f : α → β} (h : MapsTo f A B) : MapsTo f⁻¹ A B⁻¹

theorem Set.MapsTo.neg {α : Type u_2} {β : Type u_3} [InvolutiveNeg β] {A : Set α} {B : Set β} {f : α → β} (h : MapsTo f A B) : MapsTo (-f) A (-B)

theorem Set.MapsTo.div {α : Type u_2} {β : Type u_3} [Div β] {A : Set α} {B₁ B₂ : Set β} {f₁ f₂ : α → β} (h₁ : MapsTo f₁ A B₁) (h₂ : MapsTo f₂ A B₂) : MapsTo (f₁ / f₂) A (B₁ / B₂)

theorem Set.MapsTo.sub {α : Type u_2} {β : Type u_3} [Sub β] {A : Set α} {B₁ B₂ : Set β} {f₁ f₂ : α → β} (h₁ : MapsTo f₁ A B₁) (h₂ : MapsTo f₂ A B₂) : MapsTo (f₁ - f₂) A (B₁ - B₂)