We can classify relations in terms of their images.  Once again, I'll restrict myself to binary relations, since it's notationally simpler.  But these definitions could be extended to general relations.

Totality

A relation \(R\subseteq A \times B\) is left-total (or right-total) if every element on the left (right) is related to some element on the right:

• Left-total: (the following are all equivalent definitions)
  • Every element of \(A\) is related to some element of \(B\)
  • \(\overleftarrow{B}=A\)
  • For every \(a\in A\), there's some \(b\in B\) for which \(aRb\)
  • No \(a\in A\) has an empty right image / every \(a\) has a non-empty right image
  • For all \(a\in A\), \(\overrightarrow{R}(a)\neq\emptyset\)
• Right-total: (the same, but reversed)
  • Every element of \(B\) is related to some element of \(A$\)
  • \(\overrightarrow{A}=B\)
  • For every \(b\in B\), there's some \(a\in B\) for which \(aRb\)
  • No \(b\in B\) has an empty left image / every \(b\) has a non-empty left image

Uniqueness

A relation \(R\subseteq A \times B\) is left-unique (or right-unique) if every element on the right (left) relates at most one thing on the left (right):

• Left-unique: (the following are equivalent)
  • Elements of \(B\) are each related to at most one element of \(A\)
  • For all \(b\in B\), \(|\overleftarrow{R}(b)|\leq 1\)
  • For all \(b\in B\), if \(a_1 R b\) and \(a_2 R b\) then \(a_1=a_2\)
• Right-unique: (the same, but reversed)
  • Elements of \(A\) are each related to at most one element of \(B\)
  • For all \(a\in A\), \(|\overrightarrow{R}(a)|\leq 1\)
  • For all \(a\in A\), if \(a R b_1\) and \(a R b_2\) then \(b_1=b_2\)

Combination properties

Different combinations of these properties make for important special kinds of relations:

• One-to-one: both left- and right-unique
• One-to-many: left-unique and not right-unique
• Many-to-one: right-unique, but not left
• Many-to-many: neither