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This is Problem 4.20 (g) of the textbook "Reading, Writing, and Proving - A Closer Look at Mathematics", Chapter 4, 2nd ed. Springer, 2003 by Ulrich Daepp and Pamela Gorkin.

Decide whether sentence (3) is true if sentences (1) and (2) are both true. Give reasons for your answers.

Let $M$ and $n$ be real numbers.
(1) If $n > M$, then $n^2 > M^2$.
(2) We know that $n < M$.
(3) So $n^2 \le M^2$.

I am confused by the following three different ideas:

  1. (3) is false.
    Reason: Take $n = -2, M = -1$.
  2. (3) is true.
    Reason: From (1), we can conclude that $n > 0$. Together with (2), we have $0 < n < M$. Therefore, we have $n^2 < M^2$, which implies $n^2 < M^2$.
  3. We cannot decide whether (3) is true or not.
    Reason: (1) can be formulated as $n > M \to n^2 > M^2$, while (2) as simply n < M. From these two formulas, we cannot logically conclude either $n^2 \le M^2$ or $n^2 > M^2$.

Which is correct? What is your opinion?

Specifically, I am not sure whether we are allowed to make reasonings based on our knowledge about real numbers, as done in the second idea. What do you think of it?

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  • $\begingroup$ The question asks whether (2) and (3) imply (1). If using (1) and (2) you can't decide whether (3) is true, then (2) and (3) don't imply (1). $\endgroup$ – Gerry Myerson Oct 14 '17 at 8:38
  • $\begingroup$ @GerryMyerson Do you mean "the question asks whether (1) and (2) imply (3)"? I am not sure whether we can make reasonings based on our knowledge about real numbers, as done in the second idea. What do you think of it? $\endgroup$ – hengxin Oct 14 '17 at 8:43
  • $\begingroup$ Yes, sorry, whether (1) and (2) imply (3). I would say you're not meant to use what you know about the reals, but it's hard to tell without context, without having the book in front of me. $\endgroup$ – Gerry Myerson Oct 14 '17 at 8:47
  • $\begingroup$ @GerryMyerson I have copied all the content of the problem itself. In that chapter (Chapter 4), the book is giving a (quite informal) introduction to logic. You can have a flavor of it in Google Books. $\endgroup$ – hengxin Oct 14 '17 at 9:01
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    $\begingroup$ I am sure there are other interpretations as well. In short, I have no idea what is actually asked, but I believe the answer is that (1) and (2) does not imply (3) regardless. $\endgroup$ – Arthur Oct 14 '17 at 9:43
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First of all if there aren't any further conditions then the deduction rule $(1)$ is wrong. Hence if you assume that it is right you can derive any statement.

Anyway there is an easy way to actually do this: Let $n,M \in \mathbb{R}$ and the deduction rule

$$(1) \quad n>M \Longrightarrow n^2 > M^2$$

true. Then let us switch the roles of $n$ and $M$ and we obtain

$$(1)' \quad M<n \Longrightarrow M^2 < n^2$$

Now subsitute $M$ with $n$ and the other way round.

$$(1)'' \quad n<M \Longrightarrow n^2 < M^2$$

Now we use $(2)$ combined with $(1)''$ and due modus ponens we obtain $$n^2 < M^2$$ which clearly implies the weaker statement $n^2 \leq M^2$.

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  • $\begingroup$ When you switched the roles of $M$ and $n$, why did you switch the $(<,>)$ sign as well? They are just variables, it's not that any of them was meant to be on one side of the $<$ sign,right? $\endgroup$ – астон вілла олоф мэллбэрг Oct 14 '17 at 9:13
  • $\begingroup$ I just made it extra slow. First step is just the inequality from an other point of view. Second step is the actual substitution. You are right you can skip this step and go immediately to $(1)''$ $\endgroup$ – Nathanael Skrepek Oct 14 '17 at 9:14

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