Animated Solution for Mathematics - Sets and Relations: The number of non-empty equivalence relations on the set $\{1,2,3\}$ is:

The problem asks for the number of non-empty equivalence relations on the set $A=\{1,2,3\}$. The secret key to solving this is the concept of a partition.

Every equivalence relation on a set corresponds exactly to a unique partition of that set. A partition is simply a way of grouping elements into non-empty, disjoint subsets.

Instead of listing complex relations, we only need to count the number of ways to partition our set $A$.

We break the problem down into cases based on the size of the groups:

Case 1: The Universal Relation The simplest way is to put all three elements into a single, giant group. This gives us the partition $\{\{1,2,3\}\}$.

In the language of relations, this corresponds to the Universal Relation, where every element is related to every other element. This accounts for 1 valid partition.

Case 2: The Two-and-One Split We can isolate one element and group the other two together. The possible partitions are: $\{\{1\},\{2,3\}\}$ $\{\{2\},\{1,3\}\}$ * $\{\{3\},\{1,2\}\}$

This gives us 3 additional partitions.

Case 3: The Identity Relation Finally, we consider the case where we do not group any elements together. We put each element into its own separate group: $\{\{1\},\{2\},\{3\}\}$.

This corresponds to the Identity Relation, where each element is only related to itself. This is our 1 final partition.

To find the total number of equivalence relations, we sum the counts from each case:

The total number of equivalence relations on the set $A$ is 5.

For your exams, remember that the number of partitions of a set of size $n$ is given by the Bell Number, denoted as $B_n$.

The sequence of Bell numbers is:

Memorizing these values can save you significant time during competitive examinations. Every complex problem is simply a collection of beautiful, interconnected ideas.