Equivalence-Class Partition: The Set Breaks Into Coloured Blocks

Every equivalence relation on a set A has a hidden picture attached: it breaks A into equivalence classes, a collection of disjoint blocks whose union is all of A. Each block is a group of elements that are equivalent to each other; no element belongs to two blocks; no element is left out. The classes partition the set — like sorting marbles into separate bowls by colour.

This article makes the partition picture tangible.

The visual claim

Take A = \{0, 1, 2, \dots, 11\} and the relation "a and b have the same remainder when divided by n." For n = 2, this partitions A into two blocks: evens and odds. For n = 3, into three blocks by remainder modulo 3. For n = 4, four blocks. For n = 5, five blocks. The slider below scans through those four choices.

The top row shows $\{0, \dots, 11\}$ partitioned modulo $2$ into two coloured blocks: the evens $\{0, 2, 4, 6, 8, 10\}$ and the odds $\{1, 3, 5, 7, 9, 11\}$. The bottom row shows the same set partitioned modulo $3$ into three coloured blocks: remainder $0$, remainder $1$, and remainder $2$. Drag the slider to scan the modulus; the number of blocks equals $n$. For $n = 4$, there are four blocks of three elements each; for $n = 5$, five blocks with slightly uneven sizes (because $12$ is not divisible by $5$).

What the picture is showing

Three observations that hold for every equivalence relation, no matter the set:

Every element belongs to exactly one block. Pick any element of A. There is one and only one coloured block containing it. This is what "partition" means — the blocks do not overlap, and together they cover A.
Two elements are equivalent if and only if they are in the same block. The relation can be read off the partition — two numbers are congruent modulo 3 iff they are in the same colour. The partition is a complete picture of the relation.
The number of blocks equals the number of equivalence classes. For mod 2, two blocks. For mod 3, three blocks. For mod 5, five blocks. The block count is a crisp numerical property of the equivalence relation — for the mod-n relation, the number of classes is always n (provided the set is large enough to populate all remainders).

Why: reflexivity puts every element into its own class; symmetry plus transitivity then glue together all the elements linked to it. The result is a non-overlapping partition, always.

Another partition: "same first letter"

Equivalence relations don't have to be numerical. Let A = the set of Indian cities \{\text{Mumbai, Madras, Delhi, Dehradun, Kolkata, Kanpur, Bengaluru, Bhopal}\} and define a \sim b if a and b start with the same letter.

The blocks are:

M: \{\text{Mumbai, Madras}\}
D: \{\text{Delhi, Dehradun}\}
K: \{\text{Kolkata, Kanpur}\}
B: \{\text{Bengaluru, Bhopal}\}

Four blocks, one for each starting letter that appears. Every city sits in exactly one block. This is a perfectly valid equivalence relation (reflexive, symmetric, transitive), and its partition picture is just as clean as the modular one.

What changes when the relation changes

Watch the slider move from n = 2 to n = 5 and notice how the picture reshapes:

Smaller n → fewer, bigger blocks. Modulo 2: two blocks of six elements each. Modulo 3: three blocks of four elements each.
Larger n → more, smaller blocks. Modulo 5: five blocks, where two blocks have three elements and three blocks have two elements (because 12 divided by 5 leaves an uneven remainder distribution).
At n = 1 (outside our slider range, but worth knowing): one block containing everything. All integers are "equivalent" under mod 1 because everything has remainder 0.
At n = 12 (also outside the slider range for this set): twelve singleton blocks, one per element. Every element is its own class — the equivalence is the identity relation.

These two extremes — "everything is one class" and "every element is its own class" — are the boundary cases. Every other partition lies between them. There is a beautiful correspondence: the number of partitions of an n-element set is the Bell number B_n (see the going-deeper section of Relations), and they arrange themselves into a lattice from the coarsest (one block) to the finest (all singletons).

Why partitions and equivalence relations are the same thing

Every equivalence relation gives a partition (as the picture shows). Remarkably, the converse is also true: every partition gives an equivalence relation. Given any way of splitting A into non-overlapping, non-empty blocks covering all of A, you can define "a \sim b iff a and b are in the same block," and this relation is automatically reflexive (every element is in its own block), symmetric (same-block-ness doesn't care about order), and transitive (if a and b are same-block, and b and c are same-block, then a and c are same-block because "same block" is a property of the block, not of which element you start from).

This one-to-one correspondence — equivalence relations \leftrightarrow partitions — is one of the most important structural results in all of algebra. The picture in the figure is not a cute illustration; it is the relation, faithfully encoded in coloured rectangles.

The reflex

When a problem says "show that \sim is an equivalence relation and describe its classes," your workflow should become:

Verify reflexive, symmetric, transitive (that is the algebraic part).
List the classes — literally write out which elements group together.
Draw the partition picture (coloured blocks) if you have paper. On exam paper this is optional, but in your head it should be automatic.
Count the blocks. That count is the cardinality of the quotient set A / \sim, which often answers the actual question.

The partition picture is the most compact way to hold an equivalence relation in your mind. Once you see "coloured blocks," you stop memorising the definition and start seeing why the three properties are exactly what a partition needs.