Question

What can you say about the relation R on a set A if R is partial order and an equivalence relation.

Answer 1

Partial order:

Let R be a binary relation on a set A.

(i) R is reflexive if xRx for every x ∈ A.

(ii) R is anti-symmetric if for all (x, y) ∈ R and (y, x) ∈ R then x = y.

(iii) R is transitive if for all (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R.

Equivalence relation:

Let R be a binary relation on a set A.

(i) R is reflexive if xRx for every x ∈ A.

(ii) R is symmetric if for all (x, y) ∈ R then (y, x) ∈ R.

(iii) R is transitive if for all (x, y) ∈ R and (y, z) ∈ R ⇒ (x, z) ∈ R

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