6
$\begingroup$

I came across this symbol $(\bigcap)$ and I do not know what it means. I have tried to check this list but I haven't gotten any clue. I am well aware that the small $\cap$ denotes intersection in set theory. I have checked all similar questions with "What does this symbol mean" but I haven't seen an answer yet.

The symbol $(\bigcap)$ is used in a context like this

$$\big(\bigcap_{i=1}^{n} A_{ci}^{(i)} > B\big)\cap \big(\bigcap_{i=0}^{n-1} A_c^{(i)} < B\big) $$ where $A_{ci}^{(i)} > B$ and $A_c^{(i)} < B$ are events.

$\endgroup$
7
  • 5
    $\begingroup$ This symbol means the intersection of an indexed family of sets. $\bigcap_{i=1}^n X_i$ means the same as $X_1\cap X_2\cap\dots\cap X_n$. $\endgroup$ Commented May 16, 2018 at 23:49
  • 2
    $\begingroup$ That looks like an intersection to me... but usually you intersect sets I have no interpretation off the top of my head what $A^i_{ci}>B$ means as a set or if the $>$ applies to the whole intersection or not. Some context of where this expression appears would be helpful $\endgroup$
    – N8tron
    Commented May 16, 2018 at 23:49
  • 1
    $\begingroup$ Maybe $\bigcap$ is being misused here to mean the conjunction of an indexed family of statements, which should be $\bigwedge$ instead (and then the $\cap$ in your context should be $\land$). $\endgroup$ Commented May 16, 2018 at 23:51
  • $\begingroup$ @ Andreas @N8tron, how does the superscript relate to the intersection of the set family? $\endgroup$ Commented May 16, 2018 at 23:55
  • 3
    $\begingroup$ Ah. Then they are shorthand for $\{\omega\in\Omega: A_{c}^{(i)}(\omega)>B(\omega)\}$ and such. (Where $\Omega$ is the outcome set, and $A_c^{(i)}, B$ and so on are random variables.) $\endgroup$ Commented May 17, 2018 at 0:30

1 Answer 1

5
$\begingroup$

$\displaystyle\bigcap_{i=1}^n A_i$ means $A_1\cap A_2\cap A_3\cap \cdots \cap A_n,$ which is the intersection of the sets $A_1,\ldots,A_n.$

An object is a member of this set if, and only if, it is a member of every one of the sets $A_1,\ldots,A_n.$

In probability, an "event" is a set of outcomes. To say an outcome is a member of the intersection of several events is to say that it is a member of every one of them; in other words every one of those events occurs.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .