what does it mean by $\# \{ i : x_i \le t \}$?

Question

I was reading through some manuscript and wanted to try to implement some stuff mentioned in the text. I came across this equation

equation $= \# \{ i : x(i) \le t \}$

I went through a couple of references and just wanna confirm that what I understand is correct.

From what I understand the equation is saying the number of element in x that is smaller and equal to t?

Answer 1

Maybe it is better to use the form :

$\# \{ i : x_i \le t \}$.

In this way, we have a sequence $(x_i)$ [i.e. $x_0,x_1, x_2,\ldots$] and the set $\{ i : x_i \le t \}$ is the set of values of $i$ such that $x_i \le t$.

Thus, because the symbol $\#$ is often used to count the number of elements of a set, the formula is counting the number of element in $(x_i)$ that are smaller-or-equal to $t$.