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Part of Keith Devlin's Coursera MOOC on mathematical thinking requires the translation of this symbolic statement into natural language:

$$ 5 < x < 7$$

Interpretation 1: $x$ is a single unknown number located somewhere between 5 and 7 on the number line.

first interpretation

Interpretation 2: x is an interval on the number line between 5 and 7. (a line segment). second interpretation

Which interpretation is correct?

(I will define x to be a REAL NUMBER)

Conor
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    You have to define x first. Whether it is a set or a number or something else.. – Qwerty Jun 20 '16 at 15:06
  • If these are the only two interpretations offered to you, I would say there is a third and even more natural English interpretation: "$x$ is between $5$ and $7$". This is, of course, just an abbreviation of interpretation 2, but to mathematicians this carries the same meaning, the other words of interpretation 2 being implied by the context and hence unnecessary. – Lee Mosher Jun 20 '16 at 15:08
  • @Lee Mosher My intuition of interpretation two was that it was a region of the number line. "x is between 5 and 7" sounds like x is a single value like in interpretation 1? – Conor Jun 20 '16 at 15:16
  • Part of the context, presumably, is that $x$ is a variable. The general meaning of a variable is well understood in mathematics. I'll qualify this by saying that there are definitely situations where one cannot assume that "variable" is well understood. Or, maybe the most abbreviated context is simply not sufficient. One might then write "$x$ is a variable taking values between $5$ and $7$", or one might prefer some of the longer answers offered below. To me, the key thing is that one is writing for other people, and one needs to keep in mind which people one is writing for. – Lee Mosher Jun 20 '16 at 15:25
  • Dear sir how should I represent a variable (taking values between 5 and 7) in my minds eye? Should I represent it as a single point between 5 and 7? Perhaps a radio tuner that I can move between 5 and 7? What does your mental representation of a variable look like? – Conor Jun 20 '16 at 15:34

2 Answers2

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The first interpretation is correct.

For the second one, you would write

$$\{x\in \mathbb R ,\ 5<x<7\},$$

or simply

$$\{5<x<7\},$$

or (thanks to a comment) :

$$x\in(5,7).$$

E. Joseph
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I would interpret $$ 5 < x < 7 $$ as

$x$ is a number strictly between $5$ and $7$

Now, hopefully, the context should make it clear what exactly $x$ is(is it a real number, complex number, integer, ...) .

Thomas
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