I am trying to express the converse and the negation of the converse of the following statement both in plain English language and in formal language.
"For all positive real numbers $x$, there exists an integer $n$ such that $\frac{1}{n} <x.$"
Given that the universe for both $x$ and $n$ are the set of all real numbers, I have expressed the statement and its negation in formal language respectively as follow:
\begin{align*}
\forall x,(x\in \mathbb{R}^+ \rightarrow \exists n,(n\in \mathbb{Z} \wedge \frac{1}{n} < x)). \\
\exists x,(x \in \mathbb{R}^+ \wedge \forall n,(n\in \mathbb{Z} \rightarrow \frac{1}{n}\geq x)).
\end{align*}
However, I am stuck on how to express the converse and the negation of the converse of the original statement.
Here is my attempted expressions for the converse and the negation of the converse of the original statement respectively.
\begin{align*}
\exists n,((n\in \mathbb{Z} \wedge \frac{1}{n} < x) \rightarrow \forall x,(x\in \mathbb{R}^+)).\\
\forall n,((n\in \mathbb{Z} \wedge \frac{1}{n} < x) \wedge \exists x,(x \in \mathbb{R}^+)).
\end{align*}
Each of the above statements is expressed in plain English language as:
\begin{align*}
&\text{"If there exists an integer $n$ such that $\frac{1}{n} < x$, then for every $x$, $x$ is a positive real number."}\\
&\text{"Every integer $n$ has some positive real number $x$ such that $\frac{1}{n}<x$."}
\end{align*}
I am not sure whether the above expressions for the converse and the negation of the converse of the original statement both in formal language and in plain English language are correct or not.
Best Answer
The correct answer to the converse of the statement in formal language is:
\begin{align*} \forall x,(\exists n,(n\in \mathbb{Z} \wedge \frac{1}{n} < x) \rightarrow (x \in \mathbb{R}^+)). \end{align*}
which is expressed in plain English language as:
\begin{align*} \text{For every real number $x$, if there exists an integer $n$ such that $\frac{1}{n} < x$, then $x$ is positive.} \end{align*}
The correct answer to the negation of the converse of the statement in formal language is:
\begin{align*} \exists x,(\exists n,(n\in \mathbb{Z} \wedge \frac{1}{n} < x) \wedge (x \notin \mathbb{R}^+)). \end{align*}
which is expressed in plain English language as:
\begin{align*} \text{There exists a real number $x$ such that there is an integer $n$ which $\frac{1}{n} < x$ and $x$ is not positive.} \end{align*}