symmetricity and transitivity

symmetric

R : A ↔ A is symmetric if for all (a1, a2) ∈ R ⇒ (a2, a1) ∈ R

the matrix will be symmetrical about the main diagonal and the digraph will have all two-way connections

antisymmetric

R : A ↔ A is antisymmetric if for all (a1, a2) ∈ R ∧ (a2, a1) ∈ R ⇒ (a1 = a2)

(the second element cannot reflect back to the first element unless they are the same element)

the matrix will have any 1 mirrored by a zero outside of the main diagonal, in the digraph any two-way connection will be a loop

symmetricity and powers

R is symmetric if R^-1 ⊆ R

simultaneous symmetricity and antisymmetricity

R can be both symmetric and antisymmetric at the same time if only self-loops exist

simultaneous symmetricity

if R : A ↔ A is both symmetric and antisymmetric then R ⊆ IA

transitive

R : A ↔ A is transitive if ∀ a1,a2,a3 ∈ A, (a1,a2) ∈ R ∧ (a2,a3) ∈ R ⇒ (a1,a3) ∈ R in a digraph, any two nodes with a two-arc path between them will also have a direct arc between them in the same direction the matrix will have a 1 in all the same spots where the matrix of R^2 has a one

transitivity and powers

R : A ↔ A is transitive if and only if R^2 ⊆ R