Negate each of the following statements in the most informative way possible (i.e., without using the words “no” or “not”).
- There is an integer \(x\) so that \(x + y\) is odd for each integer \(y\).
- If \(x\) is an odd integer, then \(xy\) is odd for every integer \(y\).
- Given \(\varepsilon > 0\), there is \(N \in \mathbb{N}\) so that whenever \(n > N\), we have \(\left| \dfrac{2+n}{1+n} - 1 \right| < \varepsilon\).
Solution
The negation is: \[ \text{For every integer }x\text{ there is an integer }y\text{ such that }x+y\text{ is even.} \]
The negation is: \[ \text{There is an odd integer }x\text{ and an integer }y\text{ such that }xy\text{ is even.} \]
The negation is: \[ \text{There exists }\varepsilon>0\text{ such that for every }N\in\mathbb N \text{ there is }n>N\text{ with } \left|\frac{n+2}{n+1}-1\right|\ge\varepsilon. \] These follow by replacing each quantifier by its opposite and negating the final predicate.