 If \(x\) and \(y\) are two real numbers, represented by Dedekind cuts with left sets \(x_L\) and \(y_L\), and right sets \(x_R\) and \(y_R\), we can define operations of addition and multiplication of real numbers in terms of operations on the members of these sets. The left set of \(x+y\) is the set of all rational numbers that result from adding a member of \(x_L\) and a member of \(y_L\), while the right set is the set of all rational numbers that result from adding a member of \(x_R\) and a member of \(y_R\). (It takes a little work to check that every rational number is in fact in one of these two sets, but the other conditions for being a Dedekind cut are straightforward to check.) If \(x\) and \(y\) are both positive, then we can define the right set of \(x\cdot y\) as the set of all rational numbers that result from multiplying a member of \(x_R\) by a member of \(y_R\), with the left set defined as the set of all other rational numbers. (Some modifications of this definition are needed if either \(x\) or \(y\) is negative.) Subtraction and division can then be defined as the inverses of these operations, just as for the rationals.