 Now let \(A\) be a set and define the relation \(I(A,x,y)\) as follows: For \(x\) and \(y\) in \(A\), \(I(A,x,y)\) if and only if for each subset \(X\) of \(A\), either \(x\) and \(y\) are both elements of \(X\) or neither is an element of \(X\). This definition is equivalent to the more usual one identifying the identity relation on a set \(A\) with the set of ordered pairs of the form \(\langle x,x\rangle\) for \(x\) in \(A\). The present definition proves more helpful in what follows.