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.

  • 5
    $\begingroup$ This symbol means the intersection of an indexed family of sets. $\bigcap_{i=1}^n X_i$ means the same as $X_1\cap X_2\cap\dots\cap X_n$. $\endgroup$ – Andreas Blass May 16 '18 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 May 16 '18 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 $\land$). $\endgroup$ – Andreas Blass May 16 '18 at 23:51
  • $\begingroup$ @ Andreas @N8tron, how does the superscript relate to the intersection of the set family? $\endgroup$ – Abdulhameed May 16 '18 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.) $\endgroup$ – Graham Kemp May 17 '18 at 0:30

$\displaystyle\bigcap_{i=1}^n A_i$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.