# What is the symbol $\triangleq$?

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

• I've only seen it used to mean that the LHS is defined to be the RHS. Commented Oct 11, 2013 at 19:03
• It is the eject button. Commented Oct 11, 2013 at 19:19
• @copper.hat, awesome!! Commented Nov 29, 2013 at 1:32

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 $U_x$.

• +1, except that I don't agree with preferring this symbol ;) Commented Oct 11, 2013 at 20:34
• 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. Commented Nov 22, 2013 at 3:22
• +1 only because it is Brian's preferred symbol and that is a big deal! Commented Dec 7, 2013 at 8:06
• Some people might prefer $\triangleq$ because it is symmetrical. Commented Nov 26, 2016 at 0:06
• @Karlo, wouldn't symmetry be undesirable, since it is the LHS that is being defined?
– Joe
Commented Jan 18, 2021 at 16:40

The symbol means it is "equal to by definition".

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.

• You are mistaking $\overset{\scriptscriptstyle\wedge}{=}$ for $\triangleq$.
– emma
Commented Jun 5, 2018 at 10:11