# How to interpret this $(\bigcap)$ mathematical symbol?

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.

• This symbol means the intersection of an indexed family of sets. $\bigcap_{i=1}^n X_i$ means the same as $X_1\cap X_2\cap\dots\cap X_n$. – Andreas Blass May 16 '18 at 23:49
• 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 – N8tron May 16 '18 at 23:49
• 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 $\land$). – Andreas Blass May 16 '18 at 23:51
• @ Andreas @N8tron, how does the superscript relate to the intersection of the set family? – Abdulhameed May 16 '18 at 23:55
• 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.) – 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 $A_1,\ldots,A_n.$