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I have been a user of latex for the lest 10 years, and I have always wondered why to write $\equiv$ we use "equiv" if the symbol that appears is usually referred to as the congruence symbol, and to write $\cong$ we use "cong" when the symbol that appears is that of an isomorphism which is, in my opinion, much more closer to an equivalence.

I am sure that there is a very good reason, but I have always wondered what is this reason. Does any one know?

enter image description here

\documentclass{article}

\begin{document}


$ A \cong  b \equiv C$


\end{document}
David Carlisle
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Gillyweeds
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    Those are just macro names. You are always free to make your own names: \newcommand\isomorph{\cong}. Different symbols means different things in different fields. – daleif Jan 18 '24 at 11:21
  • As others wrote, they can be redefined to whatever your want. For me anything with the word isomorph or even morph needs arrows! – yannisl Jan 18 '24 at 11:45
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    Unicode calls the first (U+2245) Approximately Equal To. The symbol is often used for congruence of geometrical objects, see e.g. https://en.wikipedia.org/wiki/Congruence_(geometry). The second (2261) is called Identical To and a similar symbol U+2263 (≣) is Strickly Equivalent To. So imho the default LaTeX names are fine. – Ulrike Fischer Jan 18 '24 at 12:22

2 Answers2

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I see no elephant, and I assure you I looked very carefully.

In order to make TeX usable, symbols must get a name. The LaTeX names are based mostly on those given by Knuth. Some of them are “agnostic”, such as \otimes or \vert, others reflect common usages.

The \cong symbol is frequently used in elementary geometry to denote “congruence”, according to Hilbert's school. It is also used in higher algebra to denote isomorphism.

On the other hand, \equiv is frequently used to denote “complete identity” (not something I like, but you see f\equiv g to denote that the functions f and g are the same, that is, they take on the same value at every point of their domains, where my algebra background would dictate f=g). It is also used in connection with Gauss congruences over the integers.

Knuth made a decision, which we may not agree with. It's too late for complaining, anyway, because thousands of documents have used \cong for denoting isomorphism.

If you're not satisfied by the semantics, define your own aliases:

\NewCommandCopy{\isom}{\cong}
\NewCommandCopy{\ceq}{\equiv}
\RenewCommandCopy{\cong}{\equiv}

although I discourage from doing the last one, because you risk misunderstandings if you share your source with coauthors.

egreg
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Both seem to be possible. From Wikipedia on Congruence in geometry (all emphasis below mine):

A symbol commonly used for congruence is an equals symbol with a tilde above it, ≅, corresponding to the Unicode character 'approximately equal to' (U+2245). In the UK, the three-bar equal sign ≡ (U+2261) is sometimes used.

And about ≡:

Identity
The triple bar symbol ≡ (U+2261, LaTeX \equiv) is often used to indicate an identity, a definition (which can also be represented by U+225D ≝ EQUAL TO BY DEFINITION or U+2254 ≔ COLON EQUALS), or a congruence relation in modular arithmetic.

And further:

The triple bar or tribar, ≡, is a symbol with multiple, context-dependent meanings indicating equivalence of two different things. Its main uses are in mathematics and logic.
...
≡ is used for the higher-level metalogical notion of logical equivalence, according to which two formulas are logically equivalent when all models give them the same value
...
In number theory, it has been used beginning with Carl Friedrich Gauss (who first used it with this meaning in 1801) to mean modular congruence: a ≡ b ( mod N ) if N divides a − b

Anyway, if you don't like the names in LaTeX then you can change them, as mentioned in a comment. And it's off-topic for the site anyway :)

Marijn
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