We often find ourselves wanting to talk about what objects a given relation might relate a given object to.  For the equality relation, for example, a given object is only equal to itself.  In fact, every object is equal to itself and not other object.  However for the \(<\) relation for numbers, any given number is less than an infinite set of other numbers.

It's useful to have a notation for writing the set of things an element is related to.  For any \(R\subseteq A\times B\), that is for any relation \(R\) between elements of the sets \(A\) and \(B\), we define the images of set elements as:

That is,

• \(\overrightarrow{R}(a)\) is the set of all the things on the right related to \(a\), and
• \(\overleftarrow{R}(b)\) is the set of things related on the left to \(b\)

The images of sets of elements are:

That is,

• \(\overrightarrow{R}(X)\) is the set of everything on the right related to something in \(X\), and
• \(\overleftarrow{R}(X)\) is the set of everything on the left related to something in \(X\)

Images of \(n\)-ary relations

We can also talk about the images of \(n\)-ary relations, but I've never seen a good notation for writing it.  It's a pity because it would be useful for writing about constraint solvers.