Confused about notation ":=" versus plain old "=" Relating to sets, I find the following in a text book: 
"...the set S := {1, 2, 3}".
The book has an extensive notation appendix, but the":=" notation is not included.
What exactly does ":=" mean, and how is it different from just "=", and how is it read?  Many thanks for any help.
 A: It is common to use $:=$ when you are defining something. That way you communicate that it isn't a formula that is derived, but something that is defined. For example, I might say that the velocity of a particle is
$$
v := \frac{d}{dt} s(t)
$$
where $s(t)$ is the position. By using $:=$ I have told the reader that $v$ is defined as the derivative of the position. 
Do you have to use $:=$ every time you define something? No, it is not an uncommon notation, but many don't use it extensively.
A: This notation is borrowed from Computer science and means (in Computer science) ‘takes the value…’. I prefer the more explicit old style 
$$f'(a) \stackrel{\text{def}}{=}\mkern1mu \lim_{h\to 0} .\frac{f(a+h)-f(a)}{h}$$      
A: The "=" is equality. The case you give is definition. 
It is also sometimes used as redefinition, e.g. x := x + 1 for a programming language.
A: The notation $A=B$ means $A$ is equal to $B$.  The notation $A:=B$ means "Let $A=B$."  It means you're saying what you will mean when you write $A$.
I suspect the $\text{“}{:=}\text{''}$ notation hasn't existed for more than about a half a century, so it's brand-new.
