Types of Relations

Relations

• Relations: terminology

  • (prerequisits) set, dot product, ordered pair

  • domain

  • range

  • image

  • pre-image

  • field: domain ∪ range

• Relations: domain and range

  • heterogeneous relation, A×B

  • homogeneous relation, endorelation, S²

• Relations wrt arity

  • nullary, ()

  • unary, R may be a property, P(x), like P= "is academy award winner"

  • binary,

  • k-ary, S₁₂ⁿ (xᵢ)

• Relations wrt operations

  • converse (inverse, transpose) relation, R⁻¹

  • composition of relations, R ∘ S (or R ; S)

↠ surjection ->> ↣ injection >-> ⇹ partial equivalence relation <-|->

Universal

R = A×B ∨ R = A² ∀ab ∈ A . (a,b) ∈ R R = {(a,b)}

Empty

∀ab ∈ A . (a,b) ∉ R, R = ∅