There is exactly one set with no elements, written $\varnothing$

Depends on

• The first-order language of set theory: $\in$, $=$, formulas with parameters, and class abbreviations
• The Axiom Schema of Separation: for each formula $\varphi$, $\forall \bar p\,\forall x\,\exists y\,\forall z\,(z \in y \leftrightarrow (z \in x \wedge \varphi(z,\bar p)))$
• The Axiom of Extensionality: $\forall x\,\forall y\,(\forall z\,(z \in x \leftrightarrow z \in y) \to x = y)$

Used by

• For f : A → B with A ≠ ∅: f is injective if and only if there is g : B → A with g ∘ f = Δ_A; for A = ∅ the empty function is injective and has a left inverse if and only if B = ∅ Corollary
• There is no set y with x ∈ y ↔ ∀ s (s ∈ ∅ → x ∈ s), so ⋂ ∅ is undefined Corollary
• A function f and sets S, T with f[S ∩ T] ⊊ f[S] ∩ f[T] Counterexample
• A symmetric and transitive relation on a two-element set that is not reflexive on it Counterexample
• Relations R and S on a two-element set with S ∘ R ≠ R ∘ S Counterexample
• Sets a and b with P(a) ∪ P(b) ⊊ P(a ∪ b) Counterexample
• Sets A, B, C with (A × B) × C ≠ A × (B × C) Counterexample
• Sets with (A × C) ∪ (B × D) ⊊ (A ∪ B) × (C ∪ D) Counterexample
• ⋃_i ∈ I Aᵢ := ⋃ {Aᵢ : i ∈ I}, and ⋂_i ∈ I Aᵢ := ⋂ {Aᵢ : i ∈ I} for I ≠ ∅ Definition
• The intersection ⋂ x of a nonempty set, the binary intersection a ∩ b := ⋂{a,b}, and disjointness Definition
• (∅,∅) = {{∅}} and (∅,{∅}) = {{∅},{∅,{∅}}}, with the characterising property checked on them Example
• {∅} × {∅,{∅}} listed in full, together with the inclusion in P(P(A ∪ B)) that makes it a set Example
• ∅ is a relation on every set, is the unique equivalence relation on ∅, is a function ∅ → B for every B, is a bijection ∅ → ∅, and is not a surjection ∅ → {∅} Example
• ⋃ ⋂ (a,b) = a for all a and b, and ⋃(a,b) = {a,b}, so both coordinates are recovered from the pair as a set Example
• A four-pair relation on {∅,{∅},{∅,{∅}}} with its domain, range, field, inverse, restriction, image and self-composite computed Example
• A two-cell partition of a three-element set, the equivalence relation it induces listed pair by pair, and the quotient set recovered from it Example
• A two-to-one function on a four-element set, its kernel, the quotient, and the canonical factorisation computed explicitly Example
• For any indexed family the product ∏_i ∈ I P(Xᵢ) contains the constant function with value ∅, and ∏_i ∈ I {i} has exactly one element Example
• P(∅) = {∅}, P({∅}) = {∅,{∅}}, and P({∅,{∅}}) listed in full Example
• FALSE: (a ∖ b) ∖ c = a ∖ (b ∖ c) for all sets a, b, c False statement
• FALSE: ⋂ ∅ = ∅ False statement
• FALSE: a symmetric and transitive relation on a set A is reflexive on A, so reflexivity is redundant in the definition of an equivalence relation False statement
• ∅ ⊆ x, x ⊆ x, inclusion is transitive, and x = y if and only if x ⊆ y and y ⊆ x Lemma
• ⋃ ∅ = ∅, ⋃ {a} = a, ⋃ {a,b} = a ∪ b, ⋂ {a} = a, and ⋂ {a,b} = a ∩ b Lemma
• For a set x ≠ ∅ the collection { z : ∀ s (s ∈ x → z ∈ s) } is a set, and it does not depend on the member of x used to separate it Lemma
• For I = {∅,{∅}} with A_∅ = A and A_{∅} = B, the map f ↦ (f(∅), f({∅})) is a bijection ∏_i ∈ I Aᵢ → A × B Lemma
• If a ⊆ b then ⋃ a ⊆ ⋃ b; if in addition a ≠ ∅ then ⋂ b ⊆ ⋂ a; and c ⊆ ⋃ a for every c ∈ a, while ⋂ a ⊆ c for every c ∈ a Lemma
• The equivalence classes of an equivalence relation are nonempty, cover A, and are pairwise equal or disjoint; conversely every such cover arises from exactly one equivalence relation Lemma
• ∏_i ∈ ∅ Aᵢ = {∅}; if Aⱼ = ∅ for some j ∈ I then ∏_i ∈ I Aᵢ = ∅; and for I = {j} the evaluation f ↦ f(j) is a bijection ∏_i ∈ I Aᵢ → Aⱼ Proposition
• A × (B ∪ C) = (A × B) ∪ (A × C), A × (B ∩ C) = (A × B) ∩ (A × C), A × (B ∖ C) = (A × B) ∖ (A × C), (A ∩ B) × (C ∩ D) = (A × C) ∩ (B × D); A × B = ∅ if and only if A = ∅ or B = ∅; and for nonempty A and B, A × B ⊆ C × D if and only if A ⊆ C and B ⊆ D Proposition
• For f : A → B and S, T ⊆ A: f[S ∪ T] = f[S] ∪ f[T], f[S ∩ T] ⊆ f[S] ∩ f[T] and f[S] ∖ f[T] ⊆ f[S ∖ T]; both inclusions are equalities for all S and T if and only if f is injective Proposition
• P(∅) = {∅}; ∅ ∈ P(a) and a ∈ P(a); a ⊆ b if and only if P(a) ⊆ P(b); ⋃ P(a) = a; ⋂ P(a) = ∅; and P(a ∩ b) = P(a) ∩ P(b) while only P(a) ∪ P(b) ⊆ P(a ∪ b) holds Proposition
• Union and intersection are commutative, associative, idempotent and absorptive; a ∪ ∅ = a, a ∩ ∅ = ∅, a ∖ ∅ = a, a ∖ a = ∅; and a ⊆ b if and only if a ∪ b = b, if and only if a ∩ b = a, if and only if a ∖ b = ∅ Proposition
• The axiom ledger for this page: which of the ZFC axioms each construction and each result actually consumes Remark

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 3 results over 2 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

• B. Kaya, MATH 320 Set Theory (METU), §1.2 and Theorem 2 (standard reference, not scraped)
• Empty set (Wikipedia) (standard reference, not scraped)
• Zermelo-Fraenkel set theory (Wikipedia) (standard reference, not scraped)