What is the smallest Equivalence Relation on the set A = {1, 2, 3}?

The smallest equivalence relation on a set is the relation that contains the least number of pairs while still satisfying the properties of an equivalence relation: reflexivity, symmetry, and transitivity.

Understanding the Set

For the set A = {1, 2, 3}, we need to consider how to relate its elements.

Properties of Equivalence Relations

• Reflexivity: Every element must be related to itself.
• Symmetry: If one element is related to another, then the second must be related to the first.
• Transitivity: If one element is related to a second, and that second is related to a third, then the first must be related to the third.

The Smallest Equivalence Relation

The smallest equivalence relation on the set A is the relation that only includes the pairs that satisfy reflexivity. This means each element is only related to itself.

Relation Representation

Thus, the smallest equivalence relation can be represented as:

• (1, 1)
• (2, 2)
• (3, 3)

This relation is often referred to as the identity relation and is the simplest form of an equivalence relation on the set A.