Logic in Computer Science

Example 1. Show that every amount of change n ≥ 8 can be paid with only 3c and 5c coins. Proof: Suppose, for the sake of contradiction, that there are some values of n ≥ 8 such that it is not possible to pay n with 3c and 5c coins. Take the set of all such values. Since all of them are natural numbers, there is, by well-ordering principle, a minimal element in this set; let’s call it k. Now, consider number k − 3. There are two possibilities. First, it can be that k − 3 < 8. Since k ≥ 8 the only choices for k are 8, 9 and 10. But 8 = 3 + 5, 9 = 3 ∗ 3 and 10 = 5 ∗ 2, so in all these cases k is representable by a sum of 3s and 5s. So it should be that k − 3 ≥ 8. But then, k − 3 is not representable as a sum of 3s and 5s either (otherwise if k − 3 = 3i + 5j, then k = 3(i + 1) + 5j.) But this contradicts the fact that k was the smallest such element, given to us by the well-ordering principle. Therefore, every n ≥ 8 is representable as a sum of 3s and 5s.