MATH.6:5.1 - A relation that cannot serve as the proposed equivalence
The claim is that every reflexive, symmetric relation is transitive. On X={a,b,c}, assign the following relation; 1 means the pair is in the relation.
| R | a | b | c |
|---|---|---|---|
| a | 1 | 1 | 0 |
| b | 1 | 1 | 1 |
| c | 0 | 1 | 1 |
The diagonal establishes reflexivity. Matching entries across the diagonal establish symmetry. But a R b and b R c hold while a R c fails. This is a countermodel with the required three witnesses.
With at most two elements, every reflexive symmetric relation is transitive: in a two-link sequence x R y R z, either x=z, or one adjacent pair is an equal pair and the other link already supplies x R z. The third element makes the failure possible.
If the work needs the smallest equivalence relation containing this R, its transitive closure adds the missing pairs and gives the single class {a,b,c}. It answers whether elements are connected through R-links. A question about the original direct relation still needs the original table. For classes on which further operations must be defined, continue with MATH.2’s operation-preservation condition.