Question of style: equiv vs. equals Sometime I have trouble discerning whether an $=$ or an $\equiv$ is most appropriate.  I believe that $\equiv$ is typically used when a new definition is being introduced, rather than a statement expressing a result.  But even if that's essentially correct (the preceding sentence), I'm still not sure  when to use which... because I have seen plenty of instances where an author introduces a function or set and uses the $=$ symbol.
Consider an example in which $\equiv$ seems appropriate. Suppose I have a sequence $(a_1,q_1,\ldots,a_n,q_n)$ with some meaning.  Then I might say, "To simplify notation, let $\xi \equiv (a_1,q_1,\ldots,a_n,q_n)$,"  or, "To simplify notation, define $\xi \equiv (a_1,q_1,\ldots,a_n,q_n)$."
Now, consider an example in which $=$ seems appropriate. Suppose $X$, $Y$, and $Z$ are well-defined random variables of interest.  Then I might say, "Let $\Psi = \{ X, Y, Z \}$ be the set of random variables in our system with...." (blah blah blah...)
Now, I do not hesitate to admit that simply because I might say those sentences above, they aren't necessarily correct.  But that's why I'm asking this question.  To me, it's unclear when to use $=$ and when to use $\equiv$.
 A: Simply put, "equals" ($=$) means that the two things are the same thing.
On the other hand, "is equivalent to" ($\equiv$) doesn't require that the two things are actually the same, only that they share a certain property.
For instance, the numbers 2 and 9 are equivalent, mod 7. They are both in the same "Equivalence class" mod 7. When you perform addition and multiplication with these two numbers, the equivalence class of the result will be the same for both. However, 2 isn't the same number as 9, even in mod 7. What can be stated is that {2} = {9}, where {x} represents the equivalence class mod 7. So {2}={9}, but $2\equiv 9$.
EDIT: Another example is equivalence of triangles. Suppose that you have two triangles, $\Delta ABC$ and $\Delta DEF$, for which $|AB|=|DE|$, $|BC|=|EF|$, and $|AC|=|DF|$. Then ABC and DEF are equivalent - that is, $\Delta ABC\equiv \Delta DEF$... but they aren't the same. But the lengths of their sides are equal.
A: One convention I'm familiar with, which is used by Dijkstra et al., is that $=$ is equality in general, and $\equiv$ is equality on boolean expressions, i.e., what's often written as $\Leftrightarrow$.
So according to this convention one writes $x = 5$, and $x = 5 \equiv 5 = x$, or equivalently $(x = 5) = (5 = x)$.
