Negate the following statements without using the word “not.”
For every real number \(x\), there is a real number \(y\) such that \(|x-y|\geq2017\).
The function \(f:\mathbb R\to\mathbb R\) is continuous.
Solution
The first negation is: \[ \text{There is an }x\in\mathbb R\text{ such that }|x-y|<2017 \text{ for every }y\in\mathbb R. \]
Using the pointwise definition of continuity on \(\mathbb R\), the second negation is: \[ \exists x\in\mathbb R\ \exists\varepsilon>0\ \forall\delta>0\ \exists y\in\mathbb R: |x-y|<\delta\ \text{ and }\ |f(x)-f(y)|\geq\varepsilon. \]