Notation

A heterogeneous binary relation R between sets A and B is a triple 𝓡 = (A,B,R) and therefore defined by its domain (A), codomain (B) and the set of the ordered pairs (a,b) such that a ∈ A and b ∈ B.

• 𝓡 = (A,B,R)

• A is the domain, a ∈ A

• B is the codomain, b ∈ B

• R is the binary relation between A and B, R = {(a,b). a ∈ A ∧ b ∈ B}

• ∃a. a ∈ A ∧ ∃b. b ∈ B. (a,b) ∈ R

• { a | a ∈ (a,b) } ⊆ A

• { b | b ∈ (a,b) } ⊆ B

• ∃p ∈ R. p = (a,b). a ∈ A ∧ b ∈ B (non-empty relation)

• ¬∃p ∈ R (empty relation)

The set of the ordered pairs (a,b) ∈ R implies a direction, that is, relations are directed. The first component in a pair is an element of domain, and it is connected to the second component of the pair which is an element of the codomain.

Not all elements of domain need to be related to an element in codomain. The set of all (distinct) first components of the ordered pairs is a subset of the domain, called the pre-image?

Not all elements of codomain B need to be related to an element in domain A. The set of all (distinct) second components of the ordered pairs is a subset of the codomain, called a range (is it also the image?).