I came across this question that tells us to validate the following statement-
For every non-empty finite set X, there exists a unique antisymmetric relation on X that is both reflexive and symmetric
My validation-
Consider X={1,2,3}. Let R be a relation on X such that R={(1,1),(2,2),(3,3)}. This relation is clearly
- reflexive as (x, x) ∈ R ∀ x ∈ X
- symmetric as (x, x) ∈ R ⇒ (x, x) ∈ R
- antisymmetric as (x,y)∈R ∧ (y,x)∉R, unless x=y
My doubt is just by using this example is my deduction right that a purely reflexive relation is the only antisymmetric relation possible on X that is both reflexive as well as symmetric