 The first operation that Boole introduced was multiplication \(xy\). The standard juxtaposition notation \(xy\) for multiplication also  had a standard meaning for operators (for example, differential operators),  namely one applied \(y\) to an object and then \(x\) is applied to the result.  As pointed out by  Theodore Hailperin (1916–2014)  (1981, p. 176; and 1986, pp. 67,68),  this established notation convention handed Boole his interpretation of the multiplication \(xy\) of elective symbols as the composition of the two operators. Thus when  encountering the expression \(xy\) without a subject one was dealing with the class \(x(y(1))\), “the result being the class whose members  are both Xs and Ys”. We call this class the intersection of \(X\) and \(Y\). In LT Boole dropped the (unnecessary) use of elective  symbols and simply let \(x\), \(y\) denote classes, with \(xy\) being their  intersection. 