What terms do you consider appropriate for the relations denoted by symbols like these:
$$\Large 1.≈\qquad 2.≉\qquad 3.⪅\qquad 4.⪉$$
- The first one should be easy: “almost equal to” and “approximately equal to” are I think both clear and widely accepted. Personally I prefer “approximately (equal to)”, while Unicode calls this symbol “almost equal to”.
- The second is harder already. Personally I'd call this “not approximately equal to”. I've heard others call it “approximately not equal” (in a different context). To me, “approximately” by itself means “almost but not exactly”, so it gets me wondering how something can be almost unequal. Is it just me, or would that term confuse others as well? If it's just me being confused, does that mean the term would be acceptable, or would it still sound strange or unprofessional, even though the meaning is clear? Unicode apparently calls this “not almost equal to”, but that might be for typographic reasons.
- This one I'd call “less or approximately equal”. But could it also be called “approximately less or equal” without becoming ambiguous? Unicode says “less-than or approximate”, switching from almost equal to approximate.
- This one is hard, I think. I could call it “not greater or approximately equal”, or I could call it “less than and not approximately equal”. The latter is more in line with the typographic rendering. Would something like “approximately less than” or “approximately strictly less than” make any sense to a common audience as well? By the way, Unicode uses “less-than and not approximate” for this symbol.
To clarify: I'm using the symbols to concisely describe the relations, but it's the relations themselves I want a term for, not the symbol I'm using.
As for the precise meaning, suppose that in a given context you have some small $\varepsilon$ defined. Then you could define the relations as
\begin{alignat*}{2}
a≈b\;&:\Leftrightarrow&\;\lvert a-b\rvert&\leqq\varepsilon \\
a≉b\;&:\Leftrightarrow&\;\lvert a-b\rvert&\gt\varepsilon \\
a⪅b\;&:\Leftrightarrow&\;a-b&\leqq\varepsilon \\
a⪉b\;&:\Leftrightarrow&\;a-b&\lt-\varepsilon
\end{alignat*}
Best Answer
$\require{cancel}$ Generally speaking, if we have two relations $R$ and $S$, then we define $$a\,\cancel {R\,}\,b\iff \lnot(a\,R\,b)\\ a\,{}^R_S\,b\iff (a\,R\, b) \vee(a\,S\,b).\tag{*}\label{moi}$$
There are some exeptions, which I personally try to avoid, like $A\subsetneqq B$.
Before I go on, I must point out a crucial inconsistency in the OP. In words the OP says: "approximately by itself means almost but not exactly". At the bottom however, it then reads $a\approx b\iff \vert a-b \vert \leqq \varepsilon$ and not $\underline {a\approx b \iff 0<\vert a-b\vert \leqq \varepsilon}$. For the record, I would like to agree on the underlined definition for "$\approx$", i.e. $a\approx a$ is false.
Now, the definitions of the symbols $\approx$, $\not\approx$, $\Large ^<_\approx$ and $\Large ^<_{\not \approx}$ is completely determined by \eqref{moi}.
Now for the relations themselves, I would just use words like close to eachother, far from eachother, just below/smaller, and a lot/way smaller.
N.B. I do think you need to be more precise about how $\vert a-b\vert$ and $a-b$ relate to zero.