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Section 1.3: Functions

Example exercise

Example 8
Show that the function \(f : \mathbb{N} \to \mathbb{N}\), given by \(f(x) = 2x\), is one-one but not onto.

Solution
The function \(f\) is one-one, for \(f(x_1) = f(x_2) \Rightarrow 2x_1 = 2x_2 \Rightarrow x_1 = x_2\).
Further, \(f\) is not onto, as for \(1 \in \mathbb{N}\), there does not exist any \(x \in \mathbb{N}\) such that \(f(x) = 2x = 1\).