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 $\equiv$.

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Both are used, so you will not cause confusion if you are reasonably consistent. And both are used in other contexts. For example, $\equiv$ is used for many kinds of equivalence relation. – André Nicolas Apr 20 '13 at 6:04
I've changed the second paragraph, which said $\equiv$ twice, which to me seemed to remove the opposition and therefore the relevance of this paragraph. I may have misunderstood though, in which case please undo my change but make it more clear what this paragraph was meant to convey. – Marc van Leeuwen Apr 20 '13 at 7:13
@MarcvanLeeuwen I revised the 2nd and 3rd paragraphs. It was unclear. Hopefully now it's more clear. – synaptik Apr 20 '13 at 14:03
In my opinion the desire to use a symbol other than "$=$" in definitions is misplaced. The thing defined is certainly equal to what it is defined as, and the word "let" or "define" clearly marks this as a definition rather than as a equation involving something that already existed. The symbol "$\equiv$" has rather different uses, where plain equality might not be appropriate. By the way, in its most frequent use I pronounce it "congruent to", even though TeX calls it \equiv. – Marc van Leeuwen Apr 20 '13 at 14:44
Has anyone else besides me seen $\equiv$ to denote function equality on the entire domain? In this sense, you would write $f(x) = 4$ to denote equality for some $x$, and $f(x) \equiv 4$ to denote equality on the entire domain of $f$. – 6005 Apr 26 '13 at 13:37

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.

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Thanks. I understand your point about using $\equiv$ to denote equivalence according to some defined equivalence class. However, I often see $\equiv$ used simply to define a simpler notation for something. In those cases, no equivalence class is put forward to govern the binary $\equiv$ operator. Are such uses incorrect, technically? – synaptik Apr 20 '13 at 6:12
I would suggest that they aren't incorrect, so much as informal. Some might use "equivalent" as a way to emphasise that it's just a notational assignment, for ease of comprehension, rather than a meaningful assignment. For instance, "$K\equiv b^2-4ac$ so that we can write the roots of a quadratic as $x=\frac{-b\pm \sqrt{K}}{2a}$" - it's not formally correct to use equivalent, but it might be a little easier to understand for lay people. – Glen O Apr 20 '13 at 6:15
Thanks for the explanation. So then, one thing is for sure... there's nothing wrong with using $=$ when introducing notational assignment, if it's clear that such assignment is being introduced (e.g. by saying "define" or "let"). Right? – synaptik Apr 20 '13 at 6:17
Generally, yes, because the new notation is being defined to be something. That is, if you say "let $K=b^2-4ac$, then every time that you see $K$, you know that it's the same as $b^2-4ac$, and you can replace one with the other. On the other hand, you can't always replace 2 with 9 in mod 7 (for instance, $3^9 \not\equiv 3^2 \pmod 7$). – Glen O Apr 20 '13 at 6:30
That makes sense. – synaptik Apr 20 '13 at 6:40

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 $(x = 5) = (5 = x)$.

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