Domain and Range of a Relation:
Consider R be a relation from A to B, that is, take R ⊆ A x B. Then
Domain R = {a: a ∈ A, (a, b) ∈ R for any b ∈ B}
i.e. domain of R is the set of all the first components of the related pairs which relates to R.
Also Range R = {b: b ∈ B, (a, b) ∈ R for any a ∈ A},
i.e. range R is the set of all second component of the ordered pairs which related to R.
Therefore Dom. R ⊆ A, Range R ⊆ B.
Total Number of Distinct Relations from A to B:
Consider the set A has m components and the set B has n members. Then the product set A x B i.e. P (A x B) may have 2mn components. A x B has 2mn different type of subsets which are different relations from A to B.
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