I found this in a Computer Science pseudocode context (see page 4 of this paper).
What does this "double less than or equals to" sign mean?
4 Answers
It means exactly the same as $\le$.
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0Is there any particular reason for using this sign instead of $\le$ ? – 2012-01-11
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3It might just be the author's preference, or the journal's convention, or just because it's an old paper and there were typographical restrictions. If you start reading even older stuff with maths in it, you'll be surprised what kooky notation and language they use (at least, it's considered kooky now). – 2012-01-11
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0The "<=" notation has an equals sign in it, just as "less than or equals" says equals. The "<_" notation doesn't have an equals sign. I don't know if that's correct, but it does seem plausible. – 2012-01-11
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3@Doug: It's likely that the latter evolved from the former; the meaning of $\leqq$ and $\le$ is the same, but convention has it that $\le$ is now the most commonly used. This is probably because there's no distinction that really needs to be made. This is in contrast to, say, $\simeq$ and $\cong$ in topology, which do mean different things (respectively, homotopy equivalence and homeomorphism). – 2012-01-11
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2${\lower 1pt\lt}\atop\raise 1pt =$ apparently came first. http://jeff560.tripod.com/relation.html . I could not find any information on when the modern symbol "$\le$" was first used. – 2012-01-11
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7In short, $\leqq=\leq$. – 2012-01-11
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0You will find it in 19th century and early 20th century math books. And (it seems) some computer science guys are still using it. – 2018-03-03
It is the same old $\le$ symbol.
Mathematical notation is far from set in stone or standardised, as might appear to a beginner. Here's a sample of the variations in the inequality signs (taken from symbols-a4.pdf):
Of course, most of this is not commonly used; my guess is that they have been included only for historical purposes and for completeness.
I add a comment for the sake of completeness. As said, in general $\geqq$ means $\geq$, but in some old math texts it is possible to find the following distinction for a real vector $x$:
- "Positive" denoted as $x > 0$: all the elements of $x$ are strictly positive;
- "Semipositive" denoted as $ x \geq 0$: all the elements of $x$ are nonnegative but at least one of them is strictly positive;
- "Nonnegative" denoted as $ x \geqq 0$: all the elements of $x$ are nonnegative.
In other terms, with this notation, $x \geq 0$ implies $x \neq 0$.
In the given context, it means “less than or equal to,” just as “≤” means. However, it is coded as a separate character (not just a glyph variant), so it could be used for some other meaning. A symbol means whatever people make it mean.
Historically, it is a glyph variant, and it has been encoded as a character primarily for compatibility reasons, see Unicode Standard, pages 491–492.
According to the ISO 80000-2 standard, the character to be used is “≤” (U+2264).