What's the most used symbol for "defined to be equal to", at least in your experience (and I'm sure there are a lot of experienced people here)? Also, which one do you think is the "the most right" of them, in the sense of making the most amount of sense (no pun intended)? The ones I frequently see in literature, papers and articles on the Web are '$\equiv$', '$:=$' and '$=_{def}$'. The first one I come across a lot, though for me it's still "reserved" for modular arithmetic. The second one seems like it's come straight out of some programming language and the last one I frequently see in philosophy papers on logic and the like. What are your thoughts on this?
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5$\begingroup$ Personally, I use $:=$, mainly because $\equiv$ is used for other things and $=_{def}$ just doesn't look very pretty to me. There other ones, such as $\triangleq$. $\endgroup$– HaydenSep 24, 2014 at 19:48
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4$\begingroup$ I use $:=$, with $\stackrel{\text{def}}=$ denoting that I apply a definition to get an equality. $\equiv$ is actually "equivalent to" or "congruent to" so it is the most ambiguous. $\endgroup$– AlexRSep 24, 2014 at 19:48
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$\begingroup$ All notations make an equal amount of sense to all others; this question is just asking which you prefer aesthetically. $\endgroup$– vadim123Sep 24, 2014 at 19:49
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1$\begingroup$ @vadim123 It's still worth to point out possible ambiguities. cf $\equiv$ for a lot of other usages, among them $1 \equiv 4 \pmod 3$ $\endgroup$– AlexRSep 24, 2014 at 19:50
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1$\begingroup$ I use the symbol ":=" when I have to, but usually just writing something like, "Define/Set/Put/Write $A = k[X]/(X^2 + 3)$" works perfectly well. $\endgroup$– anomalySep 24, 2014 at 20:17
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3 Answers
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$$:=$$ is the commonest symbol to denote "is equal by definition."
Note that $$\equiv$$ is used to denote an algebraic identity: this means that the equation is true for all permitted values of its variables. Rarely, however, it may denote a definition, so it's best to use this symbol only for congruences or identities.
In short: $$:=$$ is the most widespread (presumably as it's the easiest to typeset) "by definition" symbol .
Other symbols used to denote a definition include $$\stackrel{\triangle}= \quad , \stackrel{\text{def}}= \quad, \stackrel{\cdot}= \quad .$$
Whilst there's no amibguity in the latter three symbols, you try typing \stackrel{\triangle}= every single time you make a definition, as opposed to the much-shorter :=. You'll then see why the latter of these two is most widespread in this context.
answered Sep 24, 2014 at 19:51
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2$\begingroup$ It's better than the god-awful $=_{def}$. Without getting into any debates about which is prettiest, the commonest symbol is $:= .$ $\endgroup$ Sep 24, 2014 at 19:54
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1$\begingroup$ Not everybody agrees with this. For example read the end of this page on J. Milne's website: jmilne.org/math/words.html $\endgroup$– ZavoshSep 24, 2014 at 20:07
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2$\begingroup$ What is your evidence for claiming that “$:=$ is the commonest symbol”? $\endgroup$– MJDSep 24, 2014 at 20:08
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1$\begingroup$ It's probably most common with computer scientists. $\endgroup$– ZavoshSep 24, 2014 at 20:10
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2$\begingroup$ I use
\triangleqinstead of\stackrel{\triangle}=for $\triangleq$. I am not sure if it is bare LaTeX of a part of some package, e.g., AMS. But it works here. $\endgroup$ Aug 4, 2022 at 15:36
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One advantage of $:=$ is that it's assymetric, meaning you can distinguish the thing being defined and the definition.
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$\begingroup$ I agree, but '$=_{def}$' has this property too. $\endgroup$ Sep 24, 2014 at 20:07
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1$\begingroup$ That's true, but as others have said, that is less pretty and harder to type. $\endgroup$– NishantSep 24, 2014 at 20:07
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$\begingroup$ Even though it's a bit harder to type, still, I consider '$:=$' to be quite ugly, and definitely uglier than '$=_{def}$' (read the comments under @alexqwx's answer). $\endgroup$ Sep 24, 2014 at 20:10
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$\begingroup$ Regarding the asymmetry, the object being defined is the one closer to the colon (e.g., to the left-hand side of $:=$). Very rarely, I see $=:$, where the object being defined is at the right-hand side. $\endgroup$ Dec 2 at 10:06
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All of the above. The most common one however is $ := $. The symbol $\equiv$ is usually used to denote a logical equivalence. The symbol $\stackrel{\mathrm{def}}=$ should just be exiled along with $\div$.
Ultimately, the symbol you choose is a matter of personal preference. I personally use $:=$.
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2$\begingroup$ Well, I've upvoted, but in my opinion $\div$ is very pretty (not that I would use it while actually doing math, God forbid :D) $\endgroup$ Sep 24, 2014 at 20:14
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$\begingroup$ Ah, the fables division symbol...I wonder when the last time I used that was...what's the $\LaTeX$ command for it? $\endgroup$– NishantSep 24, 2014 at 21:13
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1$\begingroup$ @Nishant, I had to search for it lol. Its \div. $\endgroup$– k170Sep 24, 2014 at 21:19
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3$\begingroup$ I have rarely seen := in a math. textbook. $\endgroup$ Sep 25, 2014 at 16:56