I know we can interpret $x\in\limsup{A_n}$ iff $x$ is in infinitely many of the $A_n$'s, but am confused as to which logical connective we need to use to describe when $x\in\limsup{A_n}$, when we write it in a sentence of symbolic logic. So is it right to say that; $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m\in\mathbb{N})(m>n\space \&\space x\in A_m)$$ or $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m\in\mathbb{N})(m>n\Rightarrow x\in A_m)$$ or $$x\in\limsup{A_n}\space\space\space\text{iff}\space\space\space(\forall n\in\mathbb{N})(\exists m>n)(x\in A_m)\space \space ?$$ So would appreciate any feedback to let me know which statement (if any) is correct and if they are all wrong please can you give me the correct statement. Thanks in advance
Asked
Active
Viewed 51 times
1
-
You can see this post. – Mauro ALLEGRANZA Mar 20 '15 at 08:27
-
The first one is the correct one, perhaps with $m \ge n$ ... – Mauro ALLEGRANZA Mar 20 '15 at 08:28
-
Thank you for your comments but to me the 1st and 3rd statements seem to say the same thing, can you help me understand why the are different? Or is it that the 3rd one is "grammatically" wrong? – Jakob Mar 20 '15 at 08:35
-
I agree ... they are the same. If we know that the "index set" is $\mathbb N$, the formula is simply : $\forall n \exists m (m \ge n \land x \in A_m)$ – Mauro ALLEGRANZA Mar 20 '15 at 08:49