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Consider the set \(\N^2 = \N \times \N\text{,}\) the set of all ordered pairs \((a,b)\) where \(a\) and \(b\) are natural numbers. Consider a function \(f: \N^2 \to \N\) given by \(f((a,b)) =a+b\text{.}\)

  1. Let \(A = \{(a,b) \in \N^2 \st a, b \le 10\}\text{.}\) Find \(f(A)\text{.}\)

  2. Find \(f\inv(3)\) and \(f\inv(\{0,1,2,3\})\text{.}\)

  3. Give geometric descriptions of \(f\inv(n)\) and \(f\inv(\{0, 1, \ldots, n\})\) for any \(n \ge 1\text{.}\)

  4. Find \(\card{f\inv(8)}\) and \(\card{f\inv(\{0,1, \ldots, 8\})}\text{.}\)

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