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\begin {ex }
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Multiplication is a function because it pairs each input---each pair
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of natural numbers---with a single output: $ \times \colon \Nat ^2 \to
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- \Nat $ . By contrast, the square root operation applied to the domain
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- $ \Nat $ is not functional, since each positive integer $ n$ has two
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- square roots: $ \sqrt {n}$ and $ -\sqrt {n}$ . We can make it functional by
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+ \Nat $ . By contrast, mapping a natural number~ $ n $ to a real~ $ x $ such
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+ that $ x^ 2 = n $ is not functional, since each positive integer~ $ n$ has two
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+ square roots: $ \sqrt {n}$ and~ $ -\sqrt {n}$ . We can make it functional by
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only returning the positive square root: $ \sqrt {\phantom {X}} \colon
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\Nat \to \Real $ .
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\end {ex }
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