CounterexampleConstruction: 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.

Relations $R$ and $S$ on a two-element set with $S \circ R \neq R \circ S$

Statement refuted

Refuted claim: $S \circ R = R \circ S$ for all relations $R$ and $S$ . With $u : = \emptyset$ , $v : = {\emptyset}$ , $A : = {u, v}$ and

$R : = {(u, v)}, S : = {(v, u)},$

the two composites are $S \circ R = {(u, u)}$ and $R \circ S = {(v, v)}$ , which are different. Composition of relations is associative ( $T \circ (S \circ R) = (T \circ S) \circ R$ , $(S \circ R)^{- 1} = R^{- 1} \circ S^{- 1}$ , $(R^{- 1})^{- 1} = R$ , $dom (R^{- 1}) = ran R$ , and $Δ_{B} \circ R = R = R \circ Δ_{A}$ for a relation $R$ from $A$ to $B$ ) but not commutative.

Facts & Assumptions

Given: $u : = \emptyset$ , $v : = {\emptyset}$ , $A : = {u, v}$ , $R : = {(u, v)}$ and $S : = {(v, u)}$ .

[L1]

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

[L2]

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 $A$ ").

[L3]

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

[L4]

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

[L5]

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

[L6]

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

[L7]

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

[L8]

$T \circ (S \circ R) = (T \circ S) \circ R$ ( $T \circ (S \circ R) = (T \circ S) \circ R$ , $(S \circ R)^{- 1} = R^{- 1} \circ S^{- 1}$ , $(R^{- 1})^{- 1} = R$ , $dom (R^{- 1}) = ran R$ , and $Δ_{B} \circ R = R = R \circ Δ_{A}$ for a relation $R$ from $A$ to $B$ ).

[L9]

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

Counterexample

technique · direct

1.1

$u \neq v$ , because $v$ has an element and $u$ has none; and $R$ and $S$ are relations on $A$ , each with one pair, both of whose coordinates lie in $A$ .

L2L5L6L9

2.1

$S \circ R$ : the only pair of $R$ is $(u, v)$ , and the only pair of $S$ whose first coordinate is $v$ is $(v, u)$ , so the only element of $S \circ R$ is $(u, u)$ .

L1L3L5L7step 1.1

2.2

$R \circ S$ : the only pair of $S$ is $(v, u)$ , and the only pair of $R$ whose first coordinate is $u$ is $(u, v)$ , so the only element of $R \circ S$ is $(v, v)$ .

L1L3L5L7step 1.1

3.1

$(u, u)$ and $(v, v)$ are different, since equality of ordered pairs would force $u = v$ ; so the two composites have different elements and $S \circ R \neq R \circ S$ , while the associativity of composition is unaffected.

L3L4L7L8step 1.1step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Composition of relations (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), Def. 12 (standard reference, not scraped)
Binary relation (Wikipedia) (standard reference, not scraped)

Relations RR and SS on a two-element set with S∘R≠R∘SS∘R=R∘S