Question: 2 In lectures, we gave a construction for the rational numbers. This started by looking at the set S of all pairs of the form

2 In lectures, we gave a construction for the rational numbers. This started by looking at the set S of all pairs of the form (a, b), with a, b E Z and b = O, and then considering the relation Q on S = Z X (Z \\ {0}) defined by: (a, b)Q[C, d) if and only if a d = b 0. Explain why things would go badly wrong if we allow 5 to include pairs (at, b) with b = O: that is, if we defined a new relation R on Z X Z by: (a, b)R(C,d) if and only if a d = b C. (Hint: the answer does not involve 'division by zero'.)

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