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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: enter image description here

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

enter image description here

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?

Rob Arthan
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Fraïssé
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  • It is not entirely clear what you are asking. There are various symbols such as those on the Wikipedia page (and also $=$ is used which is not listed there) that are used to denote a definition, that it the symbols on the left are a synonym for those on the right. The symbols are not consistently used, and when they are used, it is to make a point rather than for want of consistency. There is nothing wrong with grouping the definition as you have done above, but it is unusual and likely to get in the way of effective communication. – copper.hat Apr 23 '17 at 22:39
  • The use of $:=$ is a matter of choice. Often $=$ is used when giving a definition. I find $:=$ is used more often on the board during a talk, since it saves the trouble of writing "Define $x$ to be..." – Jair Taylor Apr 23 '17 at 22:44
  • Personally, I'd rather not use $:=$ for defining functions. After all, what we write in order to define the function $f$ is that the to be defined function has the property that for each $x$ its value $f(x)$ equals the value $\sin x$. – Hagen von Eitzen Oct 07 '17 at 14:37

2 Answers2

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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

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

Ben Grossmann
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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.

Artimis Fowl
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