I am trying to read the following two formulas in English but I am not certain how to, although I do understand their meaning.
Union:
$\bigcup_{i=1}^{n} A_{i} = \{x\, |\, \exists_i \in I(x \in A_i) \}$
The meaning of this is that the union of all sets $A_i$ is the set where a member exists in atleast one set in $A_i$.
Intersection:
$\bigcap_{i=1}^{n} A_{i} = \{x\, |\, \forall_i \in I(x \in A_i) \}$
The meaning of this is that the intersection of all sets $A_i$ is the set where a member exists in all sets in $A_i$.
For the union my best guess of how it is read is:
The union of set $A_i$ for $i\in I$ equals the set of members $x$ where there exists a set $i$ in all the sets $I$ where $x$ is a member of set $i$.
However I would like to the correct way of reading both of the formulae. Thanks.
Best Answer
Well, first you have it wrong unless you say that $I=\{1,\dots,n\}$. Then the statement simplifies:
The union of $A_i$ for $i\in I$ is the set of all such $x$ that there exists $i\in I$ with $x\in A_i$.
The intersection of $A_i$ for $i\in I$ is the set of all such $x$ that for all $i\in I$ we have $x\in A_i$.
You're almost right, just all the sets is a very unfortunate (and actually wrong) thing to say here, $I$ is a single set (of indices).