What is the distinction between $=$ and $:=$ when writing math?

Question

I am persistently confused by the usage of $=$ and $:=$ when reading papers and writing in math. Quick google search on "When to use defined as (symbol)", returned no useful result.

The way I am currently using $:=$ is basically assign the name of a variable/quantity to another. For example, let $f(x):= \sin(x)$, then clearly $f'(x) = \cos(x)$. But if I write it this way, then wouldn't it also be acceptable to write:

$$f(x):= \sin(x) := \dfrac{\exp(ix) - \exp(-ix)}{2i}$$ After all, isn't $\sin$ defined as the difference of complex exponential over $2i$?

I have seldom seen anyone write this way (certainly not in any introductory calculus textbooks), therefore I am reluctant to follow this convention.

As another example, On Wikipedia, it writes:

But it doesn't follow this convention when defining functions, for instance:

I have also seen very varied usage of $:=$ when introducing sets. Some authors prefer to say something like: let $C:= \{x\in \mathbb{R}^2|x_1^2+x_2^2 = 1\}$, other prefer just to say $C= \{x\in \mathbb{R}^2|x_1^2+x_2^2 = 1\}$

Can anyone elaborate on under which context I should use $:=$ over $=$? What is a way to keep consistency?

Answer 1

It is never wrong to use "$=$" in a definition. However, some authors like to use "$:=$" in order to signify and draw attention to the fact that the equality in question is a definition (of the object on the colon side of the equality).

For instance: if you want to define your function $f(x)$ as $f(x) = \sin(x)$, then writing $f(x) := \sin(x)$ draws attention to the fact that $f(x)$ "is defined to be" $\sin(x)$. On the other hand, that second colon seems wrong. We should not write $$ f(x) := \sin(x) := \frac{\exp(ix) - \exp(ix)}{2i} $$ unless that latter expression is what you're using for the definition of the function $x \mapsto \sin(x)$ (and of course, this is not common practice). It is perfectly fine, however, to write

$$ f(x) := \sin(x) = \frac{\exp(ix) - \exp(ix)}{2i} $$

Answer 2

I believe $:=$ is a stylistic choice for when you're introducing something new - rather than a useful identity about something previously defined or used earlier in the document. Essentially, it's an easy way to signal that $h$ in: $$h := x^* - x$$ is being defined, and so the reader need not hunt through the above text to find exactly what this variable $h$ is, and that this statement is not an assertion to be shown/having just been shown. If the context makes this clear, you needn't bother with the colon, and if you're stating something you actually want to show (like that $\sin(x) = \frac {e^{ix} - e^{-ix}} {2i}$, given the trigonometry definition of $\sin$), you should certainly not use the colon.