A relation R on a set A is called an equivalence relation if and only if it satisfies three essential properties:
1. Reflexive
2. Symmetric
3. Transitive
Let’s go step-by-step with meaning, formulas, and examples
1. Reflexive Property
A relation R on A is reflexive if (a,a)∈R ∀a∈A Meaning: Every element is related to itself.
Example:
Let A= {1,2,3} and R= {(1,1), (2,2), (3,3)}
Here, every element is related to itself ⇒ Reflexive.
2. Symmetric Property
A relation RRR is symmetric if (a,b)∈R⇒(b,a)∈R Meaning: If a is related to b, then b is also related to a.
Example:
Let A={1,2,3} and R={(1,2),(2,1)}
Since (1,2)∈R⇒(2,1)∈R(1,2) in R, relation is symmetric.
Transitive Property
A relation R is transitive if (a,b)∈R and (b,c)∈R⇒(a,c)∈R
Example:
Let R= {(1,2), (2,3), (1,3)}
Since (1,2) and (2,3) imply (1,3), the relation is transitive.
Combining all three: Equivalence Relation
A relation that is reflexive + symmetric + transitive is called an equivalence relation.