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I've come across an expression from the Stanford Neural Network Course Note that looks like this:

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(where a, b, c are parameter vectors)

What is the name of this notation "𝟙" and what does it mean / do? i.e.

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I guess it means "1" (one) but not entirely sure as it seems to be written in a specific way for a reason? (could it be that it means true if x is within the range a to b, and false if x is outside that range? Boolean related?)

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This is the indicator function. In your case,

$$ \mathbf 1(a_i < x < b_i)(x) = \begin{cases} 1 & \text{ if } a_i < x < b_i \\ 0 & \text{ else} \end{cases} $$

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It is the "indicator function". It equals one when the conditions in the subscript are met, and zero otherwise.

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From the (little) context, my guess is that $$1\!\!1(\phi(x))=\begin{cases}1&\text{if }\phi(x)\\0&\text{if }\neg\phi(x)\end{cases}$$

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