ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passverified 2026-08-06 (claude-opus-5)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A four-pair relation on ${\emptyset, {\emptyset}, {\emptyset, {\emptyset}}}$ with its domain, range, field, inverse, restriction, image and self-composite computed

Example

Write $u : = \emptyset$ , $v : = {\emptyset}$ and $w : = {\emptyset, {\emptyset}}$ ; these are pairwise distinct, since $u$ has no element, $v$ has exactly the element $u$ , and $w$ has $v$ as an element while $v$ does not. Put $A : = {u, v} \cup {w}$ and

$R : = {(u, v), (v, w)} \cup {(w, u), (u, u)} .$

Then $R$ is a relation on $A$ with

$dom R = ran R = fld R = A, R^{- 1} = {(v, u), (w, v)} \cup {(u, w), (u, u)},$

$R ↾ {u} = {(u, v), (u, u)}, R [{u} \cup {v}] = A, R \circ R = {(u, w), (u, v)} \cup {(u, u), (v, u)} \cup {(w, v), (w, u)} .$

Facts & Assumptions

Given: $u : = \emptyset$ , $v : = {\emptyset}$ , $w : = {\emptyset, {\emptyset}}$ , $A : = {u, v} \cup {w}$ and $R$ as displayed.

[L1]

A relation is a set $R$ every element of which is an ordered pair (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L2]

$dom R : = {a : \exists b (a, b) \in R}, ran R : = {b : \exists a (a, b) \in R}$ (Relation, $dom R$ , $ran R$ , $fld R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L3]

$(b, a) \in R^{- 1}$ holds if and only if $(a, b) \in R$ ; $(a, c) \in S \circ R$ holds if and only if $(a, b) \in R$ and $(b, c) \in S$ for some $b$ ; and $(a, b) \in R ↾ A$ holds if and only if $(a, b) \in R$ and $a \in A$ (The inverse relation $R^{- 1}$ , the composite $S \circ R$ , and the restriction $R ↾ A$ ).

[L4]

$b \in R [A]$ holds if and only if $(a, b) \in R$ for some $a \in A$ (The image $R [A]$ and the preimage $R^{- 1} [B]$ of a set under a relation).

[L5]

$(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ( $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$ ).

[L6]

$(a, b) : = {{a}, {a, b}}$ (The Kuratowski ordered pair $(a, b) : = {{a}, {a, b}}$ ).

[L7]

${x, y}$ is the set whose elements are exactly $x$ and $y$ , and ${x} : = {x, x}$ (The unordered pair ${x, y}$ and the singleton ${x} = {x, x}$ ).

[L8]

There is exactly one set with no elements, written $\emptyset$ (There is exactly one set with no elements, written $\emptyset$ ).

[L9]

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ( $⋃ \emptyset = \emptyset$ , $⋃ {a} = a$ , $⋃ {a, b} = a \cup b$ , $⋂ {a} = a$ , and $⋂ {a, b} = a \cap b$ ).

[L10]

$z \in A \times B$ holds if and only if $z = (a, b)$ for some $a \in A$ and some $b \in B$ (The Cartesian product $A \times B : = {z \in P (P (A \cup B)) : \exists a \in A \exists b \in B z = (a, b)}$ ).

[L11]

If every $z$ satisfies $z \in x$ if and only if $z \in y$ , then $x = y$ (The Axiom of Extensionality: $\forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \to x = y)$ ).

[L12]

$a \cup b : = ⋃ {a, b}$ (The union $⋃ x$ of a set, and the binary union $a \cup b : = ⋃ {a, b}$ ).

Verification

technique · direct

1.1

$u$ , $v$ and $w$ are pairwise distinct and $A$ has exactly them as elements; every element of $R$ is an ordered pair with both coordinates in $A$ , so $R$ is a relation on $A$ , that is $R \subseteq A \times A$ .

L1L6L7L8L9L10L12

2.1

Reading off first coordinates gives $u$ , $v$ , $w$ , so $dom R = A$ ; reading off second coordinates gives $v$ , $w$ , $u$ , so $ran R = A$ ; and the field, being the union of the two, is $A$ as well.

L2L5L9L11step 1.1

2.2

Reversing each pair gives $R^{- 1} = {(v, u), (w, v)} \cup {(u, w), (u, u)}$ ; keeping the pairs whose first coordinate is $u$ gives $R ↾ {u} = {(u, v), (u, u)}$ ; and the elements related to $u$ or to $v$ are $v$ , $u$ and $w$ , so $R [{u} \cup {v}] = A$ .

L3L4L5L7L9L11step 1.1

2.3

For the self-composite, each pair $(a, b)$ of $R$ is matched with every pair of $R$ whose first coordinate is $b$ : from $(u, v)$ and $(v, w)$ comes $(u, w)$ ; from $(u, u)$ and the two pairs starting at $u$ come $(u, v)$ and $(u, u)$ ; from $(v, w)$ and $(w, u)$ comes $(v, u)$ ; and from $(w, u)$ and the two pairs starting at $u$ come $(w, v)$ and $(w, u)$ . These six pairs are exactly the displayed $R \circ R$ .

L3L5L9L11step 1.1

3.1

The domain, range, field, inverse, restriction, image and self-composite are as displayed.

step 2.1step 2.2step 2.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

B. Kaya, MATH 320 Set Theory (METU), §2.1 (standard reference, not scraped)
Binary relation (Wikipedia) (standard reference, not scraped)
Composition of relations (Wikipedia) (standard reference, not scraped)

A four-pair relation on {∅,{∅},{∅,{∅}}}{∅,{∅},{∅,{∅}}} with its domain, range, field, inverse, restriction, image and self-composite computed