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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.

OldRoboCoder
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4 Answers4

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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.

Thomas
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This notation is borrowed from Computer science and means (in Computer science) ‘takes the value…’. I prefer the more explicit old style $$f'(a) \stackrel{\text{def}}{=}\mkern1mu \lim_{h\to 0} .\frac{f(a+h)-f(a)}{h}$$

Bernard
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  • Are you sure maths $:=$ is borrowed from computer science, and not the other way around? I prefer $:=$ regardless, for both its simplicity and because it can be turned around. – leftaroundabout Jun 25 '16 at 00:20
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    Absolutely sure. Never seen it as a student. It's programming syntax. Furthermore, it's generally not typed the right way in LaTeX. – Bernard Jun 25 '16 at 00:53
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    A new user attempted to attach the following comment as an edit to your answer: "Found usages of := to indicate definition as far back as 1894, Logica Matematica by Cesare Burali-Forti. So the notation was not necessarily borrowed from Computer Science and the early computer scientists probably borrowed it from mathematicians." – Robert Soupe Jun 25 '16 at 01:59
  • @RobertSoupe Note, for generations to come: This book is available from the Internet Archive, and I could not confirm that := is used for definitions in it. I don't speak Italian, but it seems on one page $=_{Def}$ is explained to mean definition? – Cecilia Dec 06 '23 at 18:43
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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.

mvw
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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.

Michael Hardy
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