Equivalence Relations

The following is an example of a relation that isn't an equivalence relation:

Consider the set of all the entries in a certain dictionary. Some entries represent words that are synonyms of one another.

Examples: The word $\text{bank}$ has many synonyms, some of which are $\text{fund, stock, savings, side, slope, coast, riverside}$.

Consider the relation that relates every two synonym words:

$\text{Synonyms} \;$$= \{(\text{bank}, \text{bank}), \,$$(\text{bank}, \text{savings}), \,$$(\text{savings}, \text{bank}), \,$$(\text{savings}, \text{savings}) \,$$(\text{bank}, \text{riverside}), \,$$(\text{riverside}, \text{bank}), \,$$(\text{riverside}, \text{riverside}), \,$$\dots\}$.

This relation is reflexive and symmetric, but it isn't transitive. Example: We have $(\text{savings}, \text{bank}) \in \text{Synonyms}$ and $(\text{bank}, \text{riverside}) \in \text{Synonyms}$ but $(\text{savings}, \text{riverside}) \not\in \text{Synonyms}$ because $\text{savings}$ isn't a synonym of $\text{riverside}$. Since $\text{Synonyms}$ isn't transitive, it isn't an equivalence relation.