My new favourite Unicode characters

For reasons that seemed good to me, I was leafing through The Unicode Standard, Version 3.0 — 1,040 pages of character-set goodness — when I stumbled across this pair of characters on page 520:

  • ≸ (U+2278) — neither less than nor greater than
  • ≹ (U+2279) — neither greater than nor less than

It moves my soul that both of these exist.

9 responses to “My new favourite Unicode characters

  1. lol.

    Under what sorts of logical or mathematical contexts are we likely to have a case that ” neither less than nor greater than” isn’t the same as “equal to”?

    To be clear: I’m not saying that, that can’t happen–I’m sure it certain sorts of math situations it certainly *can* happen. But what are those? The simple case would be if something were infinity or undefined. Are there any others?

    Furry cows moo and decompress.

  2. I suppose they are also not equal to… So, something for the fuzzy math / quantum dynamics boffins to appreciate? :-)

  3. Well, I can easily see why “neither less than nor greater than” is needed: if you’re dealing with complex numbers, (3+4i) is less than (6+4i) and greater than (3+2i), but neither less than nor greater than (2+5i) — nor (or course) equal to it.

    Where I really can’t make head or tail of this is when I try to imagine a situation where x is neither less than nor greater than y, but it’s not the case that x is neither greater than nor less than y.

  4. @Mike: I doubt you’d really say »less than« for any relation between complex numbers. If you do, you better state exactly the condition you intend to use.

    An example could be the real line with two copies of the 0. Both of them are all negative numbers. Yet, they are not identical and neither of them is > or < than the other (as: if it were there’d be a positive/negative number than it).

    All that said, I never encountered the sign in a maths text I read (and they do not seem to be offered by the standard LaTeX packages, so they can’t be that relevant…).

    Given that the logical meaning of both symbols should be the same, I doubt anybody will see a difference between them. It may just be a matter of the writer’s preference.

  5. They’re used for partial orderings (e.g. floating-point with NaNs), and maybe for non-strict orderings (where elements can be equivalent in the order but not equal).

    As for why there are two variants, Unicode may have just inherited them from TeX, which (of course) has both (and everything else imaginable).

    Apparently you’re not the first person to wonder about these

  6. Partially ordered sets are an important area of study with major results like Dilworth’s Theorem ( http://en.wikipedia.org/wiki/Dilworth's_theorem ). Most of the examples I see are either from graph theory or projections of multi-dimensional vectors. Mike’s complex number example is one of the second.

    Consider that you might have a set of complex numbers and need to order them somehow so you project them onto the imaginary axis. Then 5+2i is neither equal to nor greater than nor less than 17+2i.

    The existence of both the characters in OP is just a little bonbon of delight, though.

  7. One of the easiest partially ordered sets is a set, say the integers, where the subsets are ordered by set inclusion. {1} < {1,2} < {1,2,3}, but {1} ≸ {2}

  8. In intuitionistic mathematics, things can indeed be neither less than nor greater than, and yet not be equal. But likely the name is just a handy mnemonic, and one might invent a new relation ≷, and let ≹ be its negation. Similarly for ≶.

  9. Wait — I can see lots of reason why why might want ≹ (or ≸, though not both). But what the heck would ≶ mean: both less than and greater than?!

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