  It was this equational argument, that \((x + y)^2 = x + y\) implies \(xy = 0\), that  led Boole to view addition \(x + y\) as a partial operation, only defined  when \(xy = 0\), that is, when \(x\) and \(y\) are disjoint classes.  The only place where he wrote down this argument was in his unpublished notes—see  Boole: Selected Manuscripts …, 1997, edited by Ivor Grattan-Guiness and Gérard Bornet, pp. 91,92. A similar equational argument, that \((x - y)^2 = x - y\) implies \(y = xy\), led  to \(x - y\) being only defined when \(y = xy\), that is, when \(y = x \cap y\),  which is the same as \(y \subseteq x\).