8
$\begingroup$

As we know, "$A_n \text{ i.o.}$" means $A_n$ happens infinitely often, where $A_n$ is an event.

I'm not sure whether "the complement of $A_n$ happens for large $n$" is the complement of the preceding event.

Can anyone help me?

$\endgroup$
0

1 Answer 1

11
$\begingroup$

Assuming you mean "the complement of $A_n$ happens for all sufficiently large $n$," then yes, the following two events are complementary:

  1. $A_n$ happens for infinitely many $n$, and

  2. $\bar{A_n}$ happens for all sufficiently large $n$ (that is, from some point $n_0$ onward.)

Indeed, if $A_n$ happens for infinitely many $n$, then after any given point $n_0$ there will be some $n \ge n_0$ such that $A_n$ happens (and therefore $\bar{A_n}$ does not happen.)

Conversely, if $A_n$ only happens for finitely many $n$, then these $n$'s where $A_n$ happens are bounded by some $n_0$, and for all $n \ge n_0$ the complement $\bar{A_n}$ must happen.

In terms of formal logic, this argument is expressed by the equivalence of the following statements, which correspond to events $\neg$(1) and (2) above, respectively:

  1. $\neg \forall n_0\,\exists n \ge n_0\, A_n$

  2. $\exists n_0\,\forall n \ge n_0\, \neg A_n$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .