1. What an Equivalence Relation Means
An equivalence relation is a relation that behaves like equality in three ways: each element relates to itself, relations work both ways, and connections pass through a middle element. These three features allow the relation to form neat groups of linked elements.
2. The Three Required Conditions
A relation R on a set A is an equivalence relation if it satisfies all three properties:
2.1. Reflexive
(a,a) \in R
Each element must relate to itself.
2.2. Symmetric
(a,b) \in R \; \Rightarrow \; (b,a) \in R
Every connection must work in both directions.
2.3. Transitive
(a,b),(b,c) \in R \; \Rightarrow \; (a,c) \in R
If the relation connects a → b and b → c, it must also include a → c.
3. Why These Three Conditions Matter
When all three properties hold, the relation behaves like a perfect equality test. Elements that relate to each other form groups that share some common feature. Each group is an equivalence class.
4. Examples of Equivalence Relations
Some everyday examples naturally satisfy all three conditions:
4.1. Example 1 (Same Colour)
Relation: “has the same colour as”.
If object A has the same colour as B, and B has the same colour as C, then A must have the same colour as C. This relation is reflexive, symmetric, and transitive.
4.2. Example 2 (Equality)
R = \{(a,a),(b,b),(c,c),\dots\}
The equality relation is the simplest equivalence relation.
4.3. Example 3 (Same Remainder Modulo n)
On integers, define a relation:
a \sim b \quad \text{if} \quad a - b \text{ is divisible by } n
This is a classic equivalence relation used in modular arithmetic.
5. Checking If a Relation Is an Equivalence Relation
To verify whether a relation is an equivalence relation, examine all three properties. Missing even one of them means the relation is not an equivalence relation.
5.1. Illustration
Let R = \{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)\} on A = \{1,2,3\}.
- Reflexive: All (a,a) exist → OK.
- Symmetric: Every pair has its reverse → OK.
- Transitive: Chains complete properly → OK.
So R is an equivalence relation.
6. What Equivalence Relations Create
Equivalence relations divide the set into disjoint groups where all elements inside each group are related to each other. These groups are called equivalence classes, and they provide a structured way to organise elements.