What's the difference between "$≡$" and "$:=$"?

Question

When I write: $$d\zeta \equiv \omega dt$$ And: $$d\zeta := \omega dt$$ Is there any difference between the two expressions ? Are they both correct ? What's in general the difference between $\equiv$ and $:=$ ? Seems to me that both of them mean "is defined by", even though I saw $\equiv$ more often with functions and $:=$ with variables.

Answer 1

$\equiv$ is a symbol for “identically equal to”. For example:

$$\tan x\equiv\frac{\sin x}{\cos x}$$

means “the left side is always equal to the right side, no matter what the variables (in this case $x$) are”. This is a fairly old-fashioned symbol, and these days you would more often see the same thing written as:

$$\tan x=\frac{\sin x}{\cos x}\text{, for all }x\in X$$

where $X$ is explicitly given (say $X=\mathbb R\setminus\{k\pi+\pi/2\mid k\in\mathbb Z\}$), which conveys a bit more information anyways.

The $:=$ symbol means “is defined as”, and is used in definitions, to introduce a new symbol or a new notion. For example, you can use it to define the tangent of an angle:

$$\tan x:=\frac{\sin x}{\cos x}$$

This use is different from the previous one: in the previous use of $\equiv$ (or $=$) we merely claimed that the tangent of $x$ always has the same value as sine divided by cosine. Here we introduce a previously (in a given context) unknown notion of a tangent, which becomes the same as sine divided by cosine by definition.

Answer 2

Both $A \equiv B$, $A := B$, and other notation such as $A≝B$ can mean "$A$ is defined to be $B$". The main reason all of these exist is that people cannot agree on a single notation.

Sometimes these mean other things. For example:

• $a \equiv b \pmod m$ means "$a$ is congruent to $b$ modulo $m$".
• $f(x) \equiv g(x)$ may mean "$f(x)$ is equal to $g(x)$ for all $x$", which is not quite the same as a definition. (See What is the difference between "$=$" and "$\equiv$"?.)
• In the context of pseudocode, $:=$ can mean more specifically an assignment, such as "$a := a+1$" to increase the value of $a$ by $1$. (This is also used in some programming languages.)
• According to xkcd, ≝ can also mean "definitely, for sure", though I have not encountered this meaning in the literature.

Ultimately, notation can help, but words help more. If I tell you "Define $f(x) = x^2$" then I do not need to use special notation to indicate that this is a definition.