Exercise 1.2.11

Question

Form the logical negation of each claim. One trivial way to do this is to simply add ``It is not the case that...'' in front of each assertion. To make this interesting, fashion the negation into a positive statement that avoids using the word ``not'' altogether. In each case, make an intuitive guess as to whether the claim or its negation is the true statement.
  \begin{enumerate}[label=(\alph*)] 
  \item For all real numbers satisfying $a<b$, there exists an $n \in \mathbf{N}$ such that $a+1 / n<b$
  \item There exists a real number $x>0$ such that $x<1 / n$ for all $n \in \mathbf{N}$.
  \item Between every two distinct real numbers there is a rational number.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item There exist real numbers satisfying $a < b$ where $a + 1/n \ge b$ for all $n \in \mathbf N$ (false).
  \item For every real number $x > 0$ there exists an $n \in \mathbf N$ such that $x \ge 1/n$ (true).
  \item There exist two real numbers $a < b$ such that if $r < b$ then $r < a$ for all $r \in \mathbf Q$ (false).
   \end{enumerate}

Exercise 1.2.11

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Exercise 1.2.11

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