Textbooks routinely say that the relation = on a set A is the trivial equivalence relation. That label sounds dismissive — as if equality is not worth discussing. But trivial here has a technical meaning, not a judgement about importance. Equality is trivial as an equivalence relation because it splits A into the smallest possible pieces: every element is its own class, alone, unrelated to anything else. It sits at one extreme of a spectrum of equivalence relations, and calling it trivial marks that extremity.

Equality really is an equivalence relation

First, the claim you have to believe before the word trivial even makes sense. The relation R = \{(a, a) \mid a \in A\} — the set of pairs where both coordinates are the same — is an equivalence relation on A.

So equality passes all three tests. It is genuinely an equivalence relation.

What are the equivalence classes of equality?

For an equivalence relation \sim on A, the equivalence class of a is [a] = \{x \in A \mid x \sim a\}. For the relation =:

[a] = \{x \in A \mid x = a\} = \{a\}

Each equivalence class is a singleton — a one-element set containing just a itself. Nobody else is in there.

If A = \{1, 2, 3, 4\}, the four equivalence classes of equality are \{1\}, \{2\}, \{3\}, \{4\}. The partition is "each element in its own box."

Four singleton equivalence classes of equality on a four-element setA large rectangle labelled A = {1, 2, 3, 4} encloses four small rounded rectangles side by side. Each small rectangle contains a single number: 1, 2, 3, and 4 respectively. The smallest possible partition of A is shown. A = {1, 2, 3, 4} under equality 1 {1} 2 {2} 3 {3} 4 {4}
Equality on $\{1, 2, 3, 4\}$ gives four equivalence classes, each a singleton. Every element is alone in its box. This is the "finest" partition: you cannot split further without producing empty boxes.

Compare that to "has the same remainder mod 2," which splits \{1, 2, 3, 4\} into just two classes \{1, 3\} and \{2, 4\} — a coarser partition with bigger boxes.

The spectrum: finest to coarsest

Every set A has at least two equivalence relations at the extremes:

In between lie all the other equivalence relations — the "interesting" ones, with classes of sizes somewhere between 1 and |A|.

Spectrum from finest equivalence relation to coarsestA horizontal layout comparing three equivalence relations on the set {1, 2, 3, 4}. On the left labelled equality, four singleton boxes. In the middle labelled mod 2, two boxes each with two elements. On the right labelled universal, a single box with all four elements. Arrows labelled "finer" and "coarser" point left and right. = (trivial/finest) same parity universal (coarsest) 1 2 3 4 13 24 1234 finer (more classes, smaller) coarser (fewer classes, bigger) every equivalence relation on A sits on this spectrum
Three equivalence relations on $\{1, 2, 3, 4\}$ ranked by fineness. Equality (left) is the finest — $4$ classes of size $1$. "Same parity" (middle) is intermediate — $2$ classes of size $2$. The universal relation (right) is the coarsest — $1$ class of size $4$. Every equivalence on this set is one of these three or something in between.

Interactive: move along the spectrum

Slider stepping through number of equivalence classesA horizontal slider from 1 to 4 with a draggable red point. The readout shows the current setting representing how many equivalence classes the partition has. At 4 the relation is equality (finest). At 1 the relation is universal (coarsest). In between are partitions with 2 or 3 classes. 1 (coarsest) 2 3 4 (trivial/finest) drag to step through coarseness
Slide from $4$ classes (equality, trivial) down to $1$ class (universal). Every equivalence relation on a $4$-element set has between $1$ and $4$ classes, and the number is called the **rank** of the partition. Equality sits at the maximum rank — as far from "collapsing everything together" as possible.

Why the label trivial?

In mathematics, trivial usually means "the extreme case where the structure collapses into something that tells you nothing new." Examples:

Equality belongs to the same family. It satisfies every equivalence-relation axiom, but it identifies nothing beyond what was already identified. "a \sim b iff a = b" does not merge any genuinely-different elements — it is the equivalence that does no work.

Non-trivial equivalence relations are the interesting ones: "has the same remainder mod n," "points to the same set of Aadhaar records," "is in the same connected component of a graph." Each of those genuinely groups distinct elements together.

A second kind of trivial: the universal relation

Some authors also call the universal relation A \times A trivial, in the opposite sense. It lumps everything together into a single class. It satisfies the axioms but gives you only one class — so again, nothing is distinguished.

Both extremes — equality and the universal relation — are sometimes called trivial because both fail to group things in any useful way. Equality refuses to group anything. The universal relation refuses to distinguish anything. Real mathematical work happens in between.

The one-line answer

Equality is the trivial equivalence relation because every equivalence class is a singleton — each element alone. It is the smallest and finest equivalence relation on any set, the baseline against which every other equivalence is measured. It tells you nothing new about the set, which is exactly what makes it "trivial" in the mathematical sense.

Related: Relations · Equivalence Relations · Equivalence-Class Partition View · Sets — Introduction