TITLE: Appropriate Notation: $\equiv$ versus $:=$
QUESTION [31 upvotes]: 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?

REPLY [7 votes]: 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.