False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-06 (claude-opus-5)

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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.
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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

Statement

False statement. If a relation $R$ on a set $A$ is symmetric and transitive, then it is reflexive on $A$ ; consequently the reflexivity clause in the definition of an equivalence relation follows from the other two and could be dropped.

Facts & Assumptions

Given: the claim above.

[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 binary relation $\sim$ on $A$ is an equivalence relation when it is reflexive, symmetric and transitive (Equivalence relation, equivalence class, and the quotient set $A/{\sim}$ ).

[L5]

$R$ is a relation on $A$ when $R \subseteq A \times A$ (Relation, $\operatorname{dom} R$ , $\operatorname{ran} R$ , $\operatorname{fld} R$ , and the specialisations "relation from $A$ to $B$ " and "relation on $A$ ").

[L6]

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

[L7]

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

[L8]

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

[L9]

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

Refutation

technique · direct

1.1

The argument that makes the claim look right: given $a \in A$ , take any $b$ with $(a,b) \in R$ ; symmetry gives $(b,a) \in R$ , and transitivity applied to $(a,b)$ and $(b,a)$ gives $(a,a) \in R$ .

L1L2L3

1.2

The witness: put $u := \varnothing$ , $v := \{\varnothing\}$ , $A := \{u,v\}$ and $R := \{(u,u)\}$ . Here $u \neq v$ , because $v$ has an element and $u$ has none.

L8L9

2.1

The gap in step 1.1 is the phrase "take any $b$ with $(a,b) \in R$ ": no hypothesis supplies such a $b$ . Symmetry and transitivity are conditional on pairs that are already in $R$ , so they constrain $R$ only at points that $R$ relates to something, and say nothing whatever about a point of $A$ that $R$ leaves untouched.

L2L3step 1.1

2.2

$R$ is a relation on $A$ : its only element is the ordered pair $(u,u)$ , whose coordinates both lie in $A$ .

L5L6L7step 1.2

2.3

$R$ is symmetric, since its only pair is its own reversal, and transitive, since the only composable pair of its members is $(u,u)$ with $(u,u)$ , whose conclusion $(u,u) \in R$ holds.

L2L3step 1.2

3.1

$R$ is not reflexive on $A$ : $v$ is an element of $A$ and $(v,v) \notin R$ , since the only element of $R$ is $(u,u)$ and $v \neq u$ .

L1L7L8step 1.2step 2.2

4.1

The claim is therefore false, and with it the conclusion drawn from it: the reflexivity clause in the definition of an equivalence relation is not redundant, since $R$ satisfies the other two clauses on $A$ and is not an equivalence relation on $A$ .

L4step 2.1step 2.3step 3.1∎

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: 24 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)

FALSE: a symmetric and transitive relation on a set AAA is reflexive on AAA, so reflexivity is redundant in the definition of an equivalence relation