# Is there a sign for 'not less than', 'not greater than', etc.?

$880$ is not greater than $950$.
I am wondering this because there is a 'not equal to': $\not=$
Not equal to is an accepted mathematical symbol - so would this be acceptable: $\not>$?
I was searching around but I couldn't find any qualified sites that would point me in that direction.

So, I would like to know if there are symbols for, not greater, less than, less than or equal to, greater than or equal to x.

Thanks for your help and time!

• Yes $\not\gt$ is perfectly acceptable -- in fact it is probably the standard symbol, it's just that you won't see it often because $\not\gt$ is equivalent to $\le$. – user137731 May 19 '16 at 13:28
• The slash through $>$ would be OK. You could always reverse the statement and use $\le$. – Ethan Bolker May 19 '16 at 13:29
• I would add that if you are in a partial rather than a total order, $\not >$ and $\leq$ are not equivalent, and it's useful to distinguish them then. – Patrick Stevens May 19 '16 at 13:32
• To add to what @PatrickStevens says, it is the law of Trichotomy that lets you conclude that "$\not >$" is the same as "$\leq$" for a total order. Specifically, for any $a$ and $b$, exactly one of the statements "$a<b$", "$a=b$", or "$a>b$" is true. This is the case for the strict order "$<$" on the reals. +1, Patrick. – MPW May 19 '16 at 13:42
• I'd say ≯ is not equivalent to ≤. For example, 2/0 ≯ 1/0 vs 2/0 ≤ 1/0, or 2+i ≯ 1+i vs 2+i ≤ 1+i. – Gnubie May 19 '16 at 17:08

To answer the question, yes. $$a \nless b\\ a \ngtr b\\ a \nleq b\qquad a \nleqq b\qquad a \nleqslant b\\ a \ngeq b\qquad a \ngeqq b\qquad a \ngeqslant b$$ and so on for many other mathematical relations $$a \nleftarrow b\\ a \nLeftarrow b\\ A \nsupseteqq B\\ A \nvdash \phi\qquad A \nVdash \phi\\ \nexists x$$

• Woah, I didnt even know half of these existed! Thanks for your answer! – Carlos Carlsen May 19 '16 at 15:07
• Latex does not seem to have a negation for $\forall$ however – GEdgar May 19 '16 at 15:08
• @GEdgar Wouldn't it just be the typical \not\forall: $\not\forall$? That doesn't look great, and I'd much prefer $\neg\forall$, but \not should work for most of these, too. E.g., \not\leq $\not\leq$ should be pretty close, if not the same, as \nleq $\nleq$. – Joshua Taylor May 19 '16 at 15:38
• @JoshuaTaylor: The negation of true-for-all is false-for-any, hence \exists \not – Ben Voigt May 19 '16 at 19:38
• @BenVoigt I'm not sure I follow what you're trying to say. The negation of $\forall x.Px$ is simply $\neg\forall x.Px$. Since $\forall x.Px \equiv \neg\exists x.\neg Px$, we have $\neg\forall x.Px \equiv \neg\neg\exists x.\neg Px$, and (in systems with double negation elimination) we can get $\exists x.\neg Px$. That's true. But I don't really see why a priori one would formulae starting with $\neg\forall x$ or $\exists x.\neg$ over the other. – Joshua Taylor May 19 '16 at 19:55

I would probably use $850 \le 950$, as order is defined for integers.

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Nizar May 19 '16 at 14:23
• It was asked for a sign, I Iproposed using $\le$, as this was not a case where the objects were incomparable – mvw May 19 '16 at 14:26
• ..., as the integers are totally ordered. – Carsten S May 19 '16 at 16:45
• This is the correct answer for integers, where $\leq$ is much more common than $\not>$. So although the latter symbol is acceptable, and actually necessary in partially ordered set, it only makes the statement more complicated to read when used with integers. – tarulen May 20 '16 at 9:50

Equality is special in that there are two ways that two real numbers $a$ and $b$ can be not equal:

$$a>b,b>a$$

So, instead of saying $a>b \;\textrm {or}\;b>a$, we write $b\neq a$.

For the others, each negation has an existing symbol, so:

$$a\not>b \iff a\leq b,\;\,a\nleq b\iff a>b$$

etc.

But like the comments say, either is OK.

• Ok...i've now received two downvotes, without any constructive feedback, for what I consider a uncontroversial post on my part. Is there something I'm missing here, or is this just the work of random downvoter trolls? – user237392 May 20 '16 at 10:40
• I am not one of the downvoters, but I was tempted. You use the symbol $\not >$ without introducing it, for instance, and you start your answer by claiming that the trichotomy $a<b,a=b,a>b$ is a special property of equality, when in fact it is a special property of real numbers. Personally, I would save only mvw's answer among all answers and comments in this thread. All the rest seems like overly complicated and technical answers to a very basic question about real numbers. – Federico Poloni May 20 '16 at 13:41
• @FedericoPoloni I don't want this to turn into a big back and forth, but perhaps I can clarify: I'm not saying anything about trichotomy, but about the symbol itself. There are two ways two numbers can be unequal, so the negation doesn't have a clear replacement symbol apart from $\neq$. As for not introducing symbols, I was using the notation introduced by the OP themselves. I was trying to explain why we see $\neq$ a lot more often than $\not >$, for example. That is all. – user237392 May 20 '16 at 16:09
• I think that symbols such as $\nleq$ are clear enough from context. I also agree that it should be pointed out that this answer relies on specific properties of the poset of real numbers. For an arbitrary poset (which is the generality in which these sorts of symbols are defined), the symbols $\nleq$ and $>$ do not have the same meaning (though the latter implies the former). – Aaron Mazel-Gee Jul 25 '18 at 10:18

I would just like to make it clear that ≮ is NOT the same as ≥

Here is an example:

1+𝑥²≮(1+𝑥)² is clearly not the same as 1+𝑥²≥(1+𝑥)²

Think of 𝑥=-0.5 and 𝑥=2 as examples to highlight this, because although when 𝑥=-0.5, 1+𝑥²≥(1+𝑥)² but when 𝑥=2, 1+𝑥²≱(1+𝑥)²

Therefore, think of ≮ as meaning "not greater than" and ≥ meaning "more than or equal to" but remember that they are not the same!

• Welcome to MSE! Consider using MathJax to typeset math in your answers and questions. – Alexander Burstein Jan 2 '18 at 17:49
• $\not <$ and $\ge$ are the same thing. $1+x^2 \ge (1+x)^2$ doesn't mean anything until $x$ is assigned a numerical value; and when that numerical value is assigned, $\not <$ and $\ge$ mean the same thing. Whe $x=-0.5$, the LHS is $1+(-0.5)^2=1.25$, and the RHS becomes $(1+(-0.5))^2=0.25$... $1.25 \not < 0.25$ and $1.25 \ge 0.25$ mean the same thing. – Ovi Jan 2 '18 at 17:53

Saying "not less than" is different from saying "greater or equal to" because there is a chance it is not greater than and only equal to, meaning it would be false to list it as greater than if it is only possibly equal, and in any case not less than.

I would like to point out that the not less than sign would work for a theory, and the theory could later be proven to be 850 equals 850, but if greater than or equal to was used, it would convey a different message, because 850 will never be greater than 850.

It's not about the logical statement as much as the undertone of the definition's subtleties.

• "there is a chance it is not greater than and only equal to" - yes, that's part of "greater than or equal to". It's one of the two. – Jair Taylor Feb 20 '19 at 3:21