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With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in

$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$

which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's blog :

$$S(x, \alpha):= \sum_{p\le x} e(\alpha p) $$

Is it user-background dependent, or are there certain circumstances in which one is more appropriate than the other?

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    $\begingroup$ The colon-equals should be used, if at all, only for definitions. I don’t use it; I think that it’s always pretty clear from context when an equals sign is a definition and when it’s a statement. (On the very rare occasions when I use a special symbol, I use $\triangleq$.) $\endgroup$
    – Brian M. Scott
    Commented Aug 13, 2012 at 15:20
  • $\begingroup$ As the book of Walter Rudin teaches, the best idea is always to write in the correct language what you are doing. Too many symbols mean a waste of time. Use words whenever possible. $\endgroup$
    – Siminore
    Commented Aug 13, 2012 at 15:41
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    $\begingroup$ @BrianM.Scott: The problem is, sometimes you're not sure if the author is making a local definition, or just using a symbol you're not familiar with (or a previous local definition you have since forgotten) and affirming an equality, which can be quite confusing. I sometimes wish people used $:=$ more often. In any event, the symbol is also used for variable assignment in Pascal and pseudocode (and maybe others, I'm not much of a programmer). $\endgroup$
    – tomasz
    Commented Aug 13, 2012 at 16:02
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    $\begingroup$ Is there a latex symbol for 'colon-equals'? Simply typing := looks sort of bad because the colon and equals aren't vertically symmetric. $\endgroup$
    – AnonymousCoward
    Commented Aug 14, 2012 at 2:51
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    $\begingroup$ @AnonymousCoward see here: tex.stackexchange.com/questions/4216/how-to-typeset-correctly $\endgroup$
    – Moderat
    Commented Aug 14, 2012 at 12:33

7 Answers 7

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An "equality by definition" is a directed mental operation, so it is nonsymmetric to begin with. It's only natural to express such an equality by a nonsymmetric symbol such as $:=\, .\ $ Seeing a formula like $e=\lim_{n\to\infty}\left(1+{1\over n}\right)^n$ for the first time a naive reader would look for an $e$ on the foregoing pages in the hope that it would then become immediately clear why such a formula should be true.

On the other hand, symbols like $=$, $\equiv$, $\sim$ and the like stand for symmetric relations between predeclared mathematical objects or variables. The symbol $\equiv$ is used , e.g., in elementary number theory for a "weakened" equality (equality modulo some given $m$), and in analysis for a "universally valid" equality: An "identity" like $\cos^2 x+\sin^2 x\equiv1$ is not meant to define a solution set (like $x^2-5x+6=0$); instead, it is expressing the idea of "equal for all $x$ under discussion".

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    $\begingroup$ "enforced equality" - i.e., an identity as opposed to a mere equation. $\endgroup$
    – J. M. ain't a mathematician
    Commented Aug 13, 2012 at 16:10
  • $\begingroup$ I guess the latter is what people have in mind when using $\equiv$ for definitions: The name is, by definition, equivalent to the expression in all and any circumstances. $\endgroup$
    – celtschk
    Commented Aug 13, 2012 at 16:58
  • $\begingroup$ @ J.M. and @celtschk: My schoolboy's English left me alone: What I actually meant was "universal". See my edit. $\endgroup$
    – Christian Blatter
    Commented Aug 13, 2012 at 17:14
  • $\begingroup$ Which doesn't affect the statement in my comment. $\endgroup$
    – celtschk
    Commented Aug 13, 2012 at 17:19
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    $\begingroup$ I have seen $e\colon=\lim_{n\to\infty}(1+n^{-1})^n$ as a slightly more appealing variant. Here, the colon indicates what is defined without forming a new operator. This readily generalizes to definitions via other means, e.g. in logic, where you might want to say $a\colon\leftrightarrow b$ without asserting equality. $\endgroup$
    – FUZxxl
    Commented Oct 2, 2017 at 15:49
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The notation $x:= y$ is preferred as $\equiv$ has another meaning in modular arithmetic (though it is almost always clear from context as to which is meant). However, there is one big advantage to using the $:=$. That is, it is not graphically symmetric and hence allows for strings such as $$ y:= f(x) \leq g(x) =: L $$ where here we are defining both $y$ and $L$. This statement would be much more cumbersome using $\equiv$, and it would not make sense if one simply wrote $$ y \equiv f(x) \leq g(x) \equiv L. $$

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    $\begingroup$ You are defining $y$ and stating an inequality? $\endgroup$
    – john
    Commented Jun 25, 2019 at 11:04
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    $\begingroup$ Here we are stating the inequality $f(x) \leq g(x)$, and we are also defining $y:= f(x)$ and $L := g(x)$. This is the kind of thing that may come up mid calculation. Some new quantities appears via manipulation, now give it a name and continue the calculation without stopping. I would not recommend this notation for more formal writing. $\endgroup$
    – nullUser
    Commented Jun 29, 2019 at 15:42
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$x:=y$ means $x$ is defined to be $y$.

The notation $\equiv$ is also (sometimes) used to mean that, but it also have other uses such as $4\equiv0$ (mod 2).

I encountered $:=$ a lot more than $\equiv$ , and it is my personal favourite.

There is also the notation $\overset{\Delta}{=}$ to mean "equal by definition"

By the way, some people also use the notaion $x=:y$ to mean $y$ is defined to be $x$

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    $\begingroup$ Yeah, $:=$ and $=:$ have the benefit of being clearly asymmetrical, so there's no room for confusion and you can use them in reverse when needed... $\endgroup$
    – tomasz
    Commented Aug 13, 2012 at 15:59
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    $\begingroup$ @tomasz - I agree, that is one of the reasons this is my favourite. $\endgroup$
    – Belgi
    Commented Aug 13, 2012 at 16:05
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The $\equiv$ symbol has different standard meanings in different contexts:

  • Congruence in number theory, and various generalizations;
  • Geometric congruence;
  • Equality for all values of the variables, as opposed to an equation in which one seeks the values that make the equation true;
  • $x$ is defined to be $y$;
  • probably a bunch of others.

But I suspect "$:=$" is not used for anything other than definitions.

So the latter at least avoids ambiguity. But if you're reading something written by someone who doesn't see it that way, you still want to understand what is being said, so you should be aware of usage conventions that you might reasonably consider less than optimal.

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It's entirely up to the whim of the author. Other symbols that can mean the same thing are $\triangleq$ and $=_{def}$. I think that only a minority of authors use any special notation, however; the majority just use a regular equals sign.

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    $\begingroup$ Sometimes, I even see $\stackrel{\text{def}}{=}$... $\endgroup$
    – J. M. ain't a mathematician
    Commented Aug 13, 2012 at 15:54
  • $\begingroup$ I prefer that notation, as the delta equals symbol seems to not be universal, and the colon equals symbol can get munged with assignment in Maple/Pascal, but this symbol is unambiguous. $\endgroup$
    – Emily
    Commented Aug 13, 2012 at 16:03
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    $\begingroup$ But many languages use $=$ for assignment. Should I therefore avoid $=$ in equations because it could be mistaken for a C/Java/Perl/Python/... assignment? $\endgroup$
    – celtschk
    Commented Aug 13, 2012 at 17:29
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Upvoting the other answers and comments... and: conventions vary. The only way to know with reasonable confidence is from context. However, one can't know whether an author "believes in" setting context. The most important criterion may be whether or not one is tracking things well enough to reasonably infer which equalities are assignments, and which are assertions. If this seems to be an issue, likely one should back-track a bit, anyway.

In practice, assignment will be clear because the left-hand side is a single symbol (even if composed of several marks), and is appearing for the first time. The first-time-appearance criterion is obviously more easily applied if all first-time appearances are highlighted by a consistent convention (rather than being buring in-line, without emphasis or fanfare).

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This is an old question, but I'd like to give a physicist's perspective.

In physics, ≡ is commonly used to denote a definition. Most physicists are also familiar with :=, but ≡ is much more common in the physics literature.

I don't think there is any reason for this, other than historical convention and habit. Perhaps one supporting reason is that most of the other common uses of the symbol ≡ have to do with pure mathematics and are rarely encountered in physics.

So, to answer the question: it is just a convention, and it is indeed very background-dependent.

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