Is there a sign for 'not less than', 'not greater than', etc.?
I was wondering about this, just now, because I was trying to write something like:
$880$ 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!
$\endgroup$ 145 Answers
$\begingroup$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 $$ a \nleftarrow b\\ a \nLeftarrow b\\ A \nsupseteqq B\\ A \nvdash \phi\qquad A \nVdash \phi\\ \nexists x $$
$\endgroup$ 9 $\begingroup$I would probably use $850 \le 950$, as order is defined for integers.
$\endgroup$ 4 $\begingroup$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.
$\endgroup$ 4 $\begingroup$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!
$\endgroup$ 2 $\begingroup$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.
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