Set theory notation.

Question

I came across this notation

$$\bigcap _{i\in\mathbb{N} }(A_i \cup B_i)$$

while reading a paper on elementary set theory, but I'm not sure of what it means. (Perhaps the intersection of all sets $A_i \cup B_i$?)

Could anybody tell me?

Answer 1

That's right, it's as you said.

$$\bigcap_{i\in\mathbb N}(A_i\cup B_i)=\{x\mid \forall i\in\mathbb N,\,x\in A_i\cap B_i\}.$$

Answer 2

$\mathbb{N}$ is the set of indices, so you have an infinite intersection of sets that are indiced by natural numbers:

Informal:

$\bigcap_{i\in \mathbb{N}} (A_i\cup B_i)=(A_0\cup B_0)\cap (A_1\cup B_1)\cap (A_2\cup B_2)\cap\dotso$