What's the difference between "≡" ("identical to") and "≣" ("strictly equivalent to")?
"Strictly equivalent to" ("≣") is UTF character 2263, classified in the Mathematical Operators block.
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1$\begingroup$ I never heard of "≣" ("strictly equivalent to"). Where did it occur? $\endgroup$– marty cohenCommented May 20, 2018 at 3:36
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15$\begingroup$ Ditto. Never heard of the four-bar version. Unless it means you have a good Wi-Fi signal. $\endgroup$– user4894Commented May 20, 2018 at 3:37
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$\begingroup$ @martycohen It's UTF character 2263. See the link I added to my answer. $\endgroup$– GeremiaCommented May 20, 2018 at 3:46
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2$\begingroup$ this looks more like a Unicode question, rather than a math question. might be the unicode people had a slot for a character in which they didn't know what to put, or had some leftover bars to dispose of. $\endgroup$– MirkoCommented May 20, 2018 at 3:51
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3$\begingroup$ $\equiv$ and similar variations are a generic symbols used to notate an equivalence relation. $=$ is the specific equivalence relation "equals" that we are used to with sets and natural numbers and by extension is also the symbol used for equality of rational, real, complex numbers etc... For more information on the definition of a specific equivalence relation, you will need to provide information about that specific equivalence relation. There are too many to provide the definitions of every one. $\endgroup$– JMoravitzCommented May 20, 2018 at 3:52
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1 Answer
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Let's say you have two expressions $X$ and $Y$. A way of expressing that they are equivalent, or equal to each other by definition, is as follows: $$X\equiv Y.\tag1$$ However, the symbol $\equiv$ also denotes congruence, e.g. $p^2\equiv 1\pmod 6$, so using the symbol for two different circumstances can cause some confusion. Thus, to denote $(1)$, some write that $$X:= Y\quad\text{ or }\quad X\triangleq Y\quad\text{ or }\quad X\stackrel{\text{def}}{=} Y.$$ But, there is also another (but less common) variation, namely, $$X\operatorname*{\equiv}\limits^{\underline{ \ \ \ }}Y\quad\text{ or with a different typeset, }\quad X\,\require{HTML} \style{display: inline-block; transform: rotate(90deg)}{\shortparallel\shortparallel}Y.$$ The "quadruple bar" is not used to denote congruence.
This post might serve useful.
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$\begingroup$ I noticed too, this “strictly equivalent” symbol (≣). Please, can you tell more about it? $\endgroup$– Hibou57Commented Apr 29, 2020 at 14:02
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1$\begingroup$ @Hibou57 Another related Math.SE post is over here (see Doug Spoonwood's answer) which discusses about notating "equality" in propositional logic. But I first came across this "quadruple" bar on a PDF which, in hindsight, I should have attached to my answer. There's nothing too significant about it though, apart from its rare use, albeit having a somewhat intuitive meaning (to me at least). $\endgroup$– Mr PieCommented Apr 30, 2020 at 12:34