In what context are these mathematical symbols used?

I just noticed that the Unicode character set contains an entire block (256 characters) of math symbols. I've got no idea how some of them should be used. For example:

• U+22DA (⋚) and U+22DB (⋛): less-than-equal-to-or-greater-than, and greater-than-equal-to-or-less-than. Why not just use the good old equals sign = ?
• U+2295 through U+2298 (⊕⊖⊗⊘): circled + − × ⁄ symbols
• U+2235 (∿): sine wave; why do we need a character for this? does anyone use this?
• U+2268 (≨): less-than-but-not-equal-to; isn't this redundant? if x < y then isn't it always true the x ≠ y?

I got these from here (you can click "more").

Thanks!

• $\oplus$ is a special symbol, indicating "direct sum"; it is different from $+$. $\otimes$ is the tensor product, different from $\times$. Both are extremely common. The "sine wave" is not a sine wave, it's a symbol used to denote weak equivalence (often used for equivalence relations, or for stating that two functions are of the same order $\sim$. Sep 11, 2011 at 19:55
• The symbol $\lt$ is used for more than simple order symbols. For example, it is used to denote substructures (subspaces, subgroups). In those cases, you need a symbol that specifies that they are proper substructures, i.e., not equal to the original substructures. Again, the symbol is used quite a lot. I don't see your first two used much, but that doesn't mean they aren't, just like the fact that you don't see the others used somehow imply they are "useless". Sep 11, 2011 at 19:57
• Maybe you could use the first for something like: if $x ⋚ y$ then $-x ⋛ -y$ (similar to $\pm$ and $\mp$)
– TMM
Sep 11, 2011 at 20:01
• "Some math symbols seem useless"... Yes, of course, most of them are when left unexplained and without context.
– t.b.
Sep 12, 2011 at 4:04
• One book I have on group theory mentions an alternative name for ⊕ (and the like) — "hot cross plus", etc. This is highly appropriate, since the lot of them can then be referred to as bunnery operators. Badoom tish. Sep 12, 2011 at 5:05

$\oplus$ and $\otimes$ are used to denote direct sum and tensor product, which are ubiquitous in mathematics. They also have a pedagogical use as symbols for addition in an abstract abelian group resp. multiplication in an abstract group when you want to make the point that groups are much more general than addition and multiplication of real or complex numbers. $\ominus$ is occasionally used in this abstract way as well.

Your "sine wave symbol" looks to me very similar to $\sim$. Certainly that is used: $$X\sim N(\mu,\sigma^2)$$ $X$ is a normally distributed random variable with expectation $\mu$ and variance $\sigma^2$. $$X\sim \operatorname{Bin}(n,p)$$ $X$ is a binomially distributed random variable with parameters $n\in\mathbb{N}$ and $p\in[0,1]$.

$$f(x) \sim \sum_{n=-\infty}^\infty c_n e^{inx}$$ The Fourier series of $f(x)$ is that series. There is no commitment to saying the series converges to $f(x)$ (in some cases it does; in others it doesn't, and it can depend on which kind of convergence is being considered).

$$a \sim b$$ $a$ is related to $b$ (just which sort of "relation" is referred to depends on the context).

• No, ∿ (U+223F: SINE WAVE) is not a mathematical operator. It is a symbol for a sine wave, defined by the Unicode standard. You are talking about ∼ (U+223C: TILDE OPERATOR). Sep 28, 2016 at 13:12
• In general, it is important not to confuse similar-looking Unicode characters. For instance, – (U+2013: EN DASH) and − (U+2212: MINUS SIGN) are different, as are ° (U+00B0: DEGREE SIGN) and º (U+00BA: MASCULINE ORDINAL INDICATOR), and β (U+03B2: GREEK SMALL LETTER BETA) and ß (U+00DF: LATIN SMALL LETTER SHARP S). Sep 28, 2016 at 13:16
• Ah, I realise now that the OP has made a mistake. The sine wave ∿ is U+223F, but the OP wrote U+2235 which is ∵ (U+2235: BECAUSE). Sep 28, 2016 at 13:23

If $A \subset B$ are linear subspaces of a Hilbert space, $B \ominus A = \{x \in B: (x,y) = 0 \text{ for all }y \in A\}$. $\ominus$ is also used for the symmetric difference of sets.

$⋚$ and $⋛$ seem appropriate for use with partial orderings; i.e. $a⋚b$ would mean that $a$ and $b$ are comparable ($a⊥b$ is another convention for symbolising comparability).

The sine wave symbol can be used when typesetting circuit diagrams to indicate that it's an AC circuit.

$\lneqq$ exists (at least if for no other reason) for completion's sake and/or symmetry with $\subsetneqq$ and friends.

$\otimes$ is also used to denote the Kronecker product of two matrices.

Sometimes, in analyses of floating-point arithmetic (like this one), $\oplus,\ominus,\otimes,$ and $\oslash$ are used instead of $+,-,\times,$ and $/$ to indicate the inexact arithmetic (i.e. with rounding) is being performed on numbers, instead of the ideal operations.

$\sim$ is also used in the representation of asymptotic expressions, e.g. Stirling's formula here. In there, it means that even though the series given is in fact divergent, truncations of it become a better approximation of the gamma (factorial) function as the argument becomes large.