$$ \text{The number of non-empty equivalence relations on the set }\left\{1, 2, 3\right\} \text{ is :} $$

An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and transitivity. The number of equivalence relations on a set is equal to the number of partitions of that set, as each partition defines an equivalence class.

For the set $$S = \{1, 2, 3\}$$, we need to find all possible partitions. A partition divides the set into non-empty, disjoint subsets whose union is the entire set.

The possible partitions are:

1. Partition with one subset:
- The entire set: $$\{\{1, 2, 3\}\}$$
This corresponds to one equivalence relation where all elements are related to each other.

2. Partitions with two subsets:
- Subset 1: $$\{1\}$$, Subset 2: $$\{2, 3\}$$
- Subset 1: $$\{2\}$$, Subset 2: $$\{1, 3\}$$
- Subset 1: $$\{3\}$$, Subset 2: $$\{1, 2\}$$
Each of these partitions corresponds to an equivalence relation where elements in the same subset are equivalent. There are three such partitions.

3. Partition with three subsets:
- Each element in its own subset: $$\{\{1\}, \{2\}, \{3\}\}$$
This corresponds to the equivalence relation where each element is only related to itself.

Total number of partitions = 1 (one subset) + 3 (two subsets) + 1 (three subsets) = 5.

Each partition defines a unique equivalence relation. Since reflexivity requires that every element is related to itself (i.e., pairs like (1,1), (2,2), (3,3) must be present), no equivalence relation can be empty. Therefore, all 5 equivalence relations are non-empty.

The number of non-empty equivalence relations is 5.

The correct option is B. 5.