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I want to state the following:

$a>b \rightarrow x>y\\ a=b \rightarrow x=y\\ a<b \rightarrow x<y$

as short as possible. I don't like the redundancy in having 3 lines looking very equal, and thought it could be stated using a mathematical symbol (I cannot yet come up with) in a one-liner.

I have thought of something like this:

$sign(a-b) = sign(x-y)$ but it will cause some confusions for the 0-case.

MJD
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Kent Munthe Caspersen
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2 Answers2

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I think I've seen something like this in the past: $$ a \gtreqqless b \implies x \gtreqqless y $$ but you probably ought to explain to readers what it means the first time you use it in any document.

The idea (I think) is that this is more like the use of multiple coordinated $\pm$ signs than like the usual $\geq$ or $\leq$.

The Mathjax for this is \gtreqqless. There's also \lesseqqgtr, which produces $\lesseqqgtr$.

David K
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Note that if $a,b$ are symmetric and $x,y$ are symmetric, and $<$ is a strict total order (such as the comparison relation on real numbers), then your three statements are equivalent to: $\def\eq{\Leftrightarrow}$ $\def\imp{\Rightarrow}$ $\def\rr{\mathbb{R}}$

$a < b \eq x < y$.

Here by symmetry I mean that:

$( a < b \eq x < y ) \imp ( b < a \eq y < x )$.

Symmetry often holds in contexts where such kinds of comparisons make sense.

For example:

$\forall a,b \in \rr\ ( a < b \eq a^3 < b^3 )$

is equivalent to:

$\forall a,b \in \rr\ ( ( a < b \imp a^3 < b^3 ) \land ( a = b \imp a^3 = b^3 ) \land ( a > b \imp a^3 > b^3 ) )$.

user21820
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  • Is that notation common and well known? If I saw it I'd have to guess what it meant and I'd guess it meant a < b if and only if x < y, which is equivalent to a $\ge$ b if and only if x $\ge$ y but not to a > b iff x>y and a=b iff x = y. – fleablood Mar 07 '16 at 16:02
  • @fleablood: It is standard notation in logic for equivalent formulae. If you like you can use "$\leftrightarrow$" or "$\Leftrightarrow$" which are the more commonly known logical connectives. Your guess is therefore right. You missed the point about symmetry though, so I should probably clarify. Editing now. – user21820 Mar 07 '16 at 23:59
  • You have the possibility that $a > b$ and $x = y$ (and vice versa), don't you? – fleablood Mar 08 '16 at 00:03
  • @fleablood: See my edit. It's possible when there is no symmetry (in the sense I meant), but when there is symmetry it is not possible. – user21820 Mar 08 '16 at 00:05