CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-06 (claude-opus-5)

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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 symmetric and transitive relation on a two-element set that is not reflexive on it

Statement refuted

Refuted claim: a symmetric and transitive relation on a set $A$ is reflexive on $A$ . Write $u := \varnothing$ and $v := \{\varnothing\}$ , put $A := \{u,v\}$ and

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

$R$ is a relation on $A$ that is symmetric and transitive, and it is not reflexive on $A$ , because $(v,v) \notin R$ .

The failure is located exactly at the point of $A$ that $R$ does not touch: $R$ is reflexive on its own field $\{u\}$ , and symmetry and transitivity constrain $R$ only there.

Facts & Assumptions

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

[L1]

$R$ is reflexive on $A$ when $(a,a) \in R$ for every $a \in A$ (Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, and connex relations on a set).

[L2]

$R$ is symmetric when $(a,b) \in R$ implies $(b,a) \in R$ , for all $a, b \in A$ (Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, and connex relations on a set).

[L3]

$R$ is transitive when $(a,b) \in R$ and $(b,c) \in R$ imply $(a,c) \in R$ , for all $a, b, c \in A$ (Reflexive, irreflexive, symmetric, asymmetric, antisymmetric, transitive, and connex relations on a set).

[L4]

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

[L5]

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

[L6]

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

[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]

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

[L9]

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

[L10]

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

[L11]

$z \in a \cup b$ holds if and only if $z \in a$ or $z \in b$ ( $\bigcup \varnothing = \varnothing$ , $\bigcup \{a\} = a$ , $\bigcup \{a,b\} = a \cup b$ , $\bigcap \{a\} = a$ , and $\bigcap \{a,b\} = a \cap b$ ).

[L12]

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

Counterexample

technique · direct

1.1

$u \neq v$ , because $v$ has the element $u$ and $u$ has none; so $A$ has exactly the two elements $u$ and $v$ .

L6L7

2.1

$R$ is a relation on $A$ : its only element is the ordered pair $(u,u)$ , and both coordinates lie in $A$ , so $R \subseteq A \times A$ .

L4L8L10step 1.1

3.1

The three characteristic sets of $R$ are $\operatorname{dom} R = \operatorname{ran} R = \operatorname{fld} R = \{u\}$ , since $(u,u)$ is its only pair.

L5L7L9L11L12step 2.1

3.2

$R$ is symmetric: the only pair in $R$ is $(u,u)$ , whose reversal is itself. It is transitive: the only composable pair of members is $(u,u)$ with $(u,u)$ , and the required conclusion $(u,u) \in R$ holds.

L2L3L9step 2.1

4.1

$R$ is reflexive on $\operatorname{fld} R$ : the only element of $\{u\}$ is $u$ , and $(u,u) \in R$ .

L1L7step 3.1

5.1

$R$ is not reflexive on $A$ : $v \in A$ , and $(v,v) \neq (u,u)$ because $v \neq u$ , so $(v,v) \notin R$ . The failure is therefore confined to the single element of $A$ lying outside $\operatorname{fld} R$ ; symmetry and transitivity say nothing about such a point, which is exactly why they do not imply reflexivity on $A$ .

L1L9step 1.1step 2.1step 3.1step 3.2step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 11 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

Equivalence relation (Wikipedia) (standard reference, not scraped)
Binary relation (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU), §3.1 (standard reference, not scraped)