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Solutions to Exercises

1.    Answer: 2. There is strong evidence that human babies, chimpanzees, and even rats have an understanding of this problem. See S. Dehaene, The Number Sense: How the Mind Creates Mathematics, Oxford, New York, 1997, pp. 23–27, 52–56.

2.    Here is one possible way to arrive at the answers. There may be other equally simple (or even simpler) ways to solve these problems.

3.    a. Given any n-subset S of [x + n + 1], there is a largest k for which #(S ∩[x + k]) = k. Given k, we can choose S to consist of any k-element subset in ways, together with {x + k + 2,x + k + 3, …,x + n + 1}.

b.  First proof. Choose a subset of ...

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