Question

why did they use → for intersection but ∧ for union?
∩F = {x | ∀A (A∈F → x∈A}
∪F = {x | ∃A (A∈F ∧ x∈A}

I would think that ∪F = {x | ∃A (A∈F → x∈A}

Answer 1

what if both are defined using a same connective? would that change the meaning of the definitions?
∩F = {x | ∀A (A∈F → x∈A}
∪F = {x | ∃A (A∈F → x∈A}

OR

∩F = {x | ∀A (A∈F ∧ x∈A}
∪F = {x | ∃A (A∈F ∧ x∈A}

Answer 2

Well \(\ldots\to\ldots\) is just another way of saying \( \lnot \ldots \lor\ldots\)

Answer 3

So it is distinct from \(\ldots\land\ldots\)

Answer 4

Are we talking about definitions of intersection and union?

Answer 5

yeah

Answer 6

What are we intersecting \(F\) with?

Answer 7

it's the family of sets. So the union/intersection are among the elements of F, which are A

Answer 8

Oh are we doing \[
\bigcup F
\]

Answer 9

yeah

Answer 10

For the union, just because there exist a element A in F, does not necessarily mean x must be in A

Answer 11

that's why -> can not be used

Answer 12

\[
\bigcup F = \{x | \exists A (A\in F \to x\in A)\}
\]This condition would always be true, and the set would contain all elements, even if they are not in a set in \(F\).

Answer 13

\[
\bigcap F = \{x | \forall A (A\in F ∧ x\in A)\}
\]This condition would always be false, because not all sets are in \(F\), one example being \(F\) itself.

Answer 14

So these definitions would return the universal set and the empty set respectively.