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I came across this new symbol while reading a document about writing proofs, and I have never seen it before.

Mirrana
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3 Answers3

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It’s is defined to be equal to; it’s my preferred symbol, but the most common one is $:=$, and I’ve also seen $\overset{\text{def}}=$.

For each $x\in X$ there is an open nbhd $U_x$ of $x$ such that ... . Then $\mathscr{U}\triangleq\{U_x:x\in X\}$ is ...

The $\triangleq$ indicates that $\mathscr{U}$ is being defined to be $\{U_x:x\in X\}$: we are not saying that some previously defined $\mathscr{U}$ is equal to the collection of these sets $U_x$.

Brian M. Scott
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    +1, except that I don't agree with preferring this symbol ;) – Hagen von Eitzen Oct 11 '13 at 20:34
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    The $\overset{\text{def}}=$ symbol is better exactly because it doesn't lead to this question being asked as often... But the triangle-equals looks neater inline, is faster to write by hand, and is language-independent. So they're evenly matched. So let's invent a new symbol, how about $\overset{\leftrightarrow}=$ implying that LHS is a shorthand for RHS. – Evgeni Sergeev Nov 22 '13 at 03:22
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    +1 only because it is Brian's preferred symbol and that is a big deal! – ILoveMath Dec 07 '13 at 08:06
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    the best is :=. –  Dec 30 '13 at 17:35
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    Some people might prefer $\triangleq$ because it is symmetrical. – Karlo Nov 26 '16 at 00:06
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    I've always seen definitions as $\equiv$, but I like the look of $\triangleq$ – fp.monkey Jun 28 '18 at 20:29
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    @Karlo, wouldn't symmetry be undesirable, since it is the LHS that is being defined? – Joe Jan 18 '21 at 16:40
  • @Joe: The fact that it is predictably the lefthand side that is being distinguished means that symmetry of the sign is irrelevant. (And I find that it is virtually always clear from context whether something is being defined, so I see no need for a special symbol at all, save in very unusual circumstances.) – Brian M. Scott Jan 18 '21 at 17:59
  • @Joe Indeed, that can be the case, for instance when you're saying $a:=b$, although $b$ should already have been defined before and $a$ not. On the other hand, when writing $a:=b\cdot c\cdot d$, is clearer that you are defining $a$. I guess it depends and it's a personal choice. – Karlo Jan 19 '21 at 01:08

  • "... we are not saying that some previously defined $\mathscr{U}$ is equal to the collection of these sets $U_x$." $\quad$ You seem to suggest that such usage is erroneous and must be avoided. What symbol do you use instead, I wonder?

     – paperskilltrees Aug 04 '22 at 15:49
  • @paperskilltrees: I have no idea how you came to the conclusion that I was suggesting anything of the kind. I am simply pointing out the difference between $$\mathscr{U}\triangleq{U_x:x\in X},$$ which defines the symbol $\mathscr{U}$, and $$\mathscr{U}={U_x:x\in X},$$ which asserts that some previously defined set $\mathscr{U}$ is equal to the set ${U_x:,\in X}$. In most contexts I would write $$\mathscr{U}={U_x:x\in X}$$ for both of those meanings, but occasionally it is useful to distinguish definitions from statements that two previously defined things are equal. – Brian M. Scott Aug 04 '22 at 18:10
  • @BrianM.Scott I was referring to when $\triangleq$ is used to mean "is equal, by a previously made definition", just like one would write $\stackrel{(23)}{=}$ meaning that "is equal, by virtue of formula (23)". Such usage of $\triangleq$ is a reference to a previously made definition and would be a verbatim copy of it or a very trivial specialisation, e.g. $$\mathscr{U}_X := {U_x:x\in X}$$ and then $$\mathscr{U}_Y\triangleq{U_x:x\in Y},$$ where $\triangleq$ is to emphasise that the equality is a trivial consequence of a definition and that no new definition has been made. – paperskilltrees Aug 04 '22 at 20:14
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    @paperskilltrees: Ah, okay; I very much doubt that I would ever use it that way. I would use a simple equals sign and add a brief explanation if I thought that one were really necessary. – Brian M. Scott Aug 04 '22 at 23:41
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The symbol means it is "equal to by definition".

Don Larynx
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-4

It actually is defined as "estimates", and also means that the content on the two sides correlates or conforms to each other.

For further reference, see here and here, the symbol is quite often used in (German?) academic context.

Burcardo
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