Hacker News Reader

Comment by kelnos | Hacker News Reader

This led me to learn:

    ⋘ U+22D8 VERY MUCH LESS-THAN
    ⋙ U+22D9 VERY MUCH GREATER-THAN

... which for some reason made me chuckle.

View on HN

Got me curious so I just did a search for "less than" and found some more interesting ones:

≰ NEITHER LESS-THAN NOR EQUAL TO - https://www.fileformat.info/info/unicode/char/2270/index.htm

≨ LESS-THAN BUT NOT EQUAL TO - https://www.fileformat.info/info/unicode/char/2268/index.htm

⪒ GREATER-THAN ABOVE LESS-THAN ABOVE DOUBLE-LINE EQUAL - https://www.fileformat.info/info/unicode/char/2a92/index.htm

How is the ⪒ used?

I could see people using this symbol when proving for example that x = 0. Say they first show x \geq 0 and then x \leq 0. Then they've shown x ⪒ 0, and therefore x = 0. But yeah, a symbol like '⪒' for this purpose is maybe natural to write on a blackboard, maybe not so much in electronic form.

I'm not aware of any other purposes.

If a ≤ b or b ≤ a. That does not always have to be the case. For example, x ≤ y could mean that x is a descendant of y in a tree.

>LESS-THAN BUT NOT EQUAL TO

what's the point of this? This is already implied with just "<"

"<" is used to represent partial-order relations, and partial orders are often non-strict (i.e. really less-than-or-equal). Then LESS-THAN BUT NOT EQUAL TO can be used to denote that a particular pair is partially-ordered but not equal (i.e. strict partially ordered). In the same vein there’s also SUBSET OF WITH NOT EQUAL TO (U+228A).

I don't understand. "equal" and "less than" are disjoint. "subset" and "equals" are not disjoint (equal sets are each others subsets), so that's a whole different example.

In mathematics, the symbol "<" is also used for non-strict partial orders, where it effectively means “less than or equal to” instead of “strictly less than”. Working with non-strict partial orders is often easier than working with the equivalent strict partial order, and for economy of notation/ease of writing some mathematicians still use “<“ then instead of “≤”. In the rare cases they want to express strictly-less-than for a particular pair in the non-strict partial order, they can use the “less than but not equals” sign.

> "subset" and "equals" are not disjoint (equal sets are each others subsets)

There is “⊂” vs. “⊆” just like there’s “<“ vs. “≤”. A set is not a strict subset of itself, merely a non-strict subset of itself.

Thanks, now I understand.

> and for economy of notation/ease of writing some mathematicians still use “<“ then instead of “≤”. In the rare cases they want to express strictly-less-than for a particular pair in the non-strict partial order, they can use the “less than but not equals” sign.

This is pure bullshit of the highest grade. Is is a perfect example of newspeak. "The symbols I use mean what I say they mean whenever I say so". It is gatekeeping. It is part of what makes math impenetrable for a lot of people. It is lazy. It is absurd. Such a lack of consistency is absurd for a rigorous field like math. It is a perfect example of my belief that mathematicians are abysmal linguists. I do not understand why mathematicians have this attitude towards their communication medium. If I could ask a genie for a wish, I would force the use of a compiler and a linter on all mathematicians. If your paper does not pass compilation and linting it gets automatically deleted. Bullshit!

/rant

Interestingly, that last one seems to be missing from LaTeX.

now I want the also the absurdist VERY MUCH LESS-THAN OR EQUAL

Oh God yes

Would that be an implicit inclusive or (i.e. very much less than, much less than, less than, or equal), or exclusive (very much less than or equal, but not much less than or less than)?

Exclusive, I reckon.

  0 ⋘===8

I want "much equal to", and "very much equal to". Which, in scientific terms, or floating point comparisons, would probably make some sense

This triggers a doge meme in my mind... Wow, such very much less-than!

When you're trying to convince STEM professionals to go from hand written design documents to computer generated, you'd better be able to reproduce every little symbol that they're accustomed to using, or else it's just an excuse for non-compliance.

Something like:

(X+1)/X ≈ 1 for X ⋙ 1

We actually use a bit for showing derivation work on engineering design documents, especially for stating requirements for an equation to be valid. Or for equations that need to be simplified to be algebraically solved.

A one line equation might have two pages of definitions, assumptions, and constraints.

In this case, ⋙ is used to mean not just much greater-than, but big enough to justify whatever the simplification it is that you're making.

Eg, 10 could be considered much greater (≫) than 1, but is still a 10% error in the equation above.

This is getting ridiculous. How long until we have things uncountably more than other things?

Could you combining-character an aleph-1 symbol on top of one of these?

In math, when you care about the details of how small one thing is compared to some other thing, you probably use little-o notation. It's the same thing you're probably familiar with from computer science.

So if y is small compared to x, you might say y = o(x), or if it's really small compared to x, you might say y = o(x²), or if it's really really really small, o(x⁶)... whatever you need for your proof.

(o(x²) is more impressively small than o(x) because x < 1)

Nice

That reminds me of something about the Greek letters Omega (Ω, ω) and Omicron (Ο, ο). O-mega means "big O" and O-micron means "small O".

Isn't that +∞ and -∞?

It seems closer to the aleph numbers.

Some infinities are countable.

Yes, but the GP's idea extends to larger infinities.

Reply on news.ycombinator.com