Set theory notation for intersection

Question

I've started reading Probability by Blikzstein and Wang and run into the following formula:

$B = \bigcap_{j=1}^{10} A_{j}$

I wasn't able to find a definition of the notation in the book, or on Wikipedia:

• https://en.wikipedia.org/wiki/Set_notation
• https://en.wikipedia.org/wiki/Set-builder_notation
• https://en.wikipedia.org/wiki/Set_theory

I think I can guess what the notation means, but I was hoping to find a formal definition.

I have seen a similar question here: Set Theory Notation Crises, however that question does not have the digits above the intersection symbol.

Answer 1

In set theory given a class term $X$ its intersection is defined as $$ \bigcap X := \{x \mid \forall y \in X, x \in y\}. $$ Now, note that we can equivalently write $$ \bigcap_{j = 1}^{10} A_j = \bigcap \{A_j \mid \forall j \in \{1, \ldots, 10\}\}. $$ Therefore $$ \bigcap_{j = 1}^{10} A_j = \{x \mid \forall j \in \{1, \ldots, 10\}, x \in A_j\}. $$ Check out 'Introduction to set theory' by Hrbacek and Jech for further info.

Answer 2

It's a short-hand notation for $B= A_1\cap A_2\cap \ldots \cap A_{10}$.