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I've encountered several usages of $:=$, for example here

$$\alpha_i := \begin{cases} x_i & i \leq n \\ A & n< i \leq 2^k \end{cases}$$

and here

Since $f$ is not constant, there exist some $a,b \in A$, $a \neq b$ such that $f(a) \neq f(b)$. Take $g$ to be $g(x):=b$ for every $x \in A$. Now $(f \circ g) (a)=f(b)$. Hence we have $(f \circ g) (a)=f(b) \neq f(a)$, so $f \circ g \neq f$.

However, I don't understand the difference between $=$ and $:=$. Is there any? I checked this source, but it didn't help.

Stokolos Ilya
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  • Typically used in the definition of functions you are using. – Andrew Chin Nov 24 '19 at 06:24
  • @AndrewChin So if I understand you correctly, there is no substantial difference between the two? – Stokolos Ilya Nov 24 '19 at 06:25
  • As far as I know, you are correct. – Andrew Chin Nov 24 '19 at 06:26
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    $:=$ stands for 'equal to by definition'. – Kavi Rama Murthy Nov 24 '19 at 06:37
  • @KaboMurphy The source I quoted within the post also said that $:=$ stood for "equal by definition", but unfortunately, I don't understand what makes it different from $=$. – Stokolos Ilya Nov 24 '19 at 06:45
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    You can use $:=$ when you are first introducing a new symbol to define what it is. e.g., $f(x) := x^2$ means that we are defining $f(x)$ to be $x^2$, not deducing that $f(x) = x^2$ from other propositions. It is often used in blackboard talks because it is faster than writing a complete sentence definition. It's generally only ever used once per object defined, and only when the symbol is first seen. – Jair Taylor Nov 24 '19 at 06:46
  • There's no functional difference, if that's what you're asking. But it can make texts easier to read since it makes it clearer that the equality is something that is taken to be true by definition, not something derived or deduced. – YiFan Tey Nov 24 '19 at 08:50

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