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In what context should I use $=$ and $\equiv$?

What is the precise difference?


(I wasn't sure what to tag this with, any suggestions?)

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(Notation) is fine, I guess. – Stijn Mar 26 '11 at 17:04
You need to give some background. What $=$ means, should be clear, and $\equiv$ can have a lot of different meanings. – Stefan Mar 26 '11 at 17:09
up vote 10 down vote accepted

The $\equiv$ symbol originally meant "is identically equal to", and as that name implies it is used with identities. It is actually stating that the equality holds for all instantiations of the free variables. For example $\sin\left(\theta+\frac{\pi}{2}\right)=\cos{\theta}$ is true for any value of $\theta$, therefore $\sin\left(\theta+\frac{\pi}{2}\right)\equiv\cos{\theta}$.

People often got that confused with "equal by definition" or "defined to be". There are separate symbols for those meanings, including $\triangleq$ and ≝ (Unicode 0x225d). The $\equiv$ symbol has been used for this purpose so often that this is now sometimes considered a correct usage.

The $\equiv$ symbol was also repurposed to mean a congruence relationship like several of the other answers have discussed.

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I don't recall seeing ≡ used to mean "equal by definition" at all. – Mariano Suárez-Alvarez Mar 27 '11 at 0:19
@Mariano, "equal by definition" is basically another way of saying "defined to be". These days the congruence relation meaning appears to be the most common. $\equiv$ is actually fairly rare these days outside of congurence relations, but the next most common usage is "defined to be". – Kevin Cathcart Mar 28 '11 at 3:44
it probably is... my comment only reflected my not ever recalling seeing it used in that way (and I do read math all day :) ) – Mariano Suárez-Alvarez Mar 28 '11 at 4:04
I'm used to seeing $:=$ to mean equal by definition – Daniel Freedman Dec 30 '11 at 13:49
While $:=$ is sometimes used to mean "equal by definition", it is also used in impure contexts like computer science to mean assignment. Since in pure mathmatics re-assignment is not possible (if $x=y$ is true now, it is always true (i.e. $(x=y)\implies(x=y)$)) in pure mathatics assignment is not distinct from defintion. But be warned that in other contexts asignment and definition are very distinct. – Kevin Cathcart Dec 30 '11 at 19:43

Use $=$ when you precisely mean that the two expressions refer to the same thing. For example, $2=1+1$ is the definition of 2, so the two sides really are the same thing. However, in advanced mathematics, people sometimes blur the use of $=$ to include isomorphic objects, e.g. $\mathbb{Z}$ and $\pi_1(\mathbb{S}^1)$, which in my opinion is terrible.

There are a few situations in which people use $\equiv$. Generally, $\equiv$ is one common notation for an equivalence relation, most often when the equivalence relation is on a ring $R$, and $a\equiv b$ when $a-b\in I$ for some ideal $I$. We then say "$a\equiv b\bmod I$". The other scenario I can think of is when we want to say that a function takes a certain value everywhere in a set, e.g. if the function $f(x)$ equals 1 for every $x\in [0,1]$, but might do something else for other inputs, I can write "$f\equiv 1$ on $[0,1]$".

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There is a more or less canonical choice of isomorphism $\mathbb Z\to\pi_1(S^1)$, the one which maps $1$ to the class of identity map, so that particular example is not that terrible, really :) – Mariano Suárez-Alvarez Mar 27 '11 at 0:18

It seems that $a \equiv b$ means that $a$ is equivalent to $b$ with respect to some equivalence relation $R$.

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Sometimes $\equiv$ is used to mean "defined to be" although I think := is more common for that.

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Consider Fermat's little theorem: $\rm\ a^p\ \equiv\ a\ \ (mod\ p)\ $ for all $\rm\ a,\ p\in \mathbb Z\:,\:\ p\:$ prime. This congruence can also be written as an equality in the ring $\rm\:\mathbb Z/p\:,\: $ e.g. $\:$ as $\rm\ \bar a^{\:p}\ =\ \bar a\ $ in $\rm\:\mathbb Z/p\:,\:$ where $\rm\:\bar a\:$ denotes the equivalence class $\rm\ a + p\ \mathbb Z\ $ of all integers congruent to $\rm\:a\:$ modulo $\rm\:p\:.\:$ Further, abusing notation, one often drops the overbar from the notation. This has the important consequence that equations in congruence rings look precisely the same as equations for integers, so that we can reuse our well-practiced intuition manipulating integer equations (valid since congruences are equivalence relations enjoying the same properties as integers equations - they can be added, multiplied, etc).

Thus, while technically, there is an important distinction between a congruence and an equality - one which is important to keep in mind when first learning about congruences - in practice this distinction is often profitably blurred so that the analogy between equalities and congruences can be exploited to the hilt. For some examples see some of my prior posts where I explicitly emphasize such points.

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