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How to write $b$ between $a$ and $c$ formally ? I mean it could be

1) $a<b<c$

or

2) $a>b>c$

but I want to leave it in the middle which one it is.

If I use the sandwich theorem for instance I think a notation for this might be usefull.

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Not sure if there is a standard notation for this. Making one up would only force your readers to try and parse what you came up with. $b$ between $a$ and $c$ sounds reasonable in itself. –  Aryabhata Apr 11 '13 at 23:53

3 Answers 3

up vote 1 down vote accepted

$$\min\{ a,c \} <b < \max\{ a,c \}$$

or $$(b-c)(b-a) <0 \,.$$

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1  
Or maybe simplest: $b\in (a,c) \cup (c,a)$.... –  N. S. Apr 11 '13 at 22:49
  • You can write that there exists $t \in (0,1)$ such that $b = ta + (1-t)c$.
  • You can write "let $I(a,c) = (a,c)$ if $a<c$ or $(c,a)$ otherwise, and $b \in I(a,c)$".
  • You can write $\min\{a,c\} < b < \max\{a,c\}$.
  • If you like, you can just write $b \in (a,c)$, borrowing the notion (from the plane of dimension 2 or more) that $(a,c)$ is the interval between the plane point $a$ and the plane point $c$ (ignoring direction).
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$a<b<c$ and $a>b>c$ are not the same (I assume that was a typo).

But, $a<b<c$ and $c>b>a$ are the same things. In this case, both notations work. The first one looks cooler and easier to understand (to me). By the way, Also, you need to make sure that the inequalities are strict, e.g. $1<1<1$ is false, but $1 \leq 1 \leq 1$ is true.

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That is his problem actually: the fact that they are not the same. $b$ between a and c, could mean either of them, so if you write one you are missing the second.... –  N. S. Apr 11 '13 at 22:48
    
-1 Your assumption is wrong, as is obvious from careful reading: "I want to leave it in the middle which one it is. If I use the sandwich theorem for instance ...". IOW, b is "sandwiched" between a and c, and the OP wants a notation that says that without specifying the order of a and c. –  Jim Balter Aug 14 at 20:12

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