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As the question title suggests, how do I visualize$$\cap_{i = 1}^\infty A_i = (\cup_{i = 1}^\infty A_i^c)^c?$$Let's start with the right-hand side. So I have a bunch of circles representing the $A_i$'s, right? Then I take the complements of each one, so for each $A_i$, the corresponding $A_i^c$ is the entire ambient space with the $A_i$ removed. I have trouble visualizing that taking the union of all those $A_i^c$'s then taking the complement of that union is our desired intersection $\cap_{i = 1}^\infty A_i$. Could anybody help me visualize this?

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    $\begingroup$ how do you even "visualize" $\cap_{i=1}^{\infty} A_i$? :P $\endgroup$
    – user290300
    Commented Aug 23, 2016 at 14:15
  • $\begingroup$ I think it's easier to consider the equivalent $$ \left( \bigcap_{i=1}^\infty A_i \right)^c = \bigcup_{i=1}^\infty A_i^c $$ $\endgroup$ Commented Aug 23, 2016 at 14:20
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    $\begingroup$ Also, I would tend not to "visualize" this at all; to me, this comes down to logical definitions. In particular, I would prove it as is done here. $\endgroup$ Commented Aug 23, 2016 at 14:23
  • $\begingroup$ In my opinion, don't visualize an infinite set. Reduce this to just two sets and think about the Venn Diagram. $\endgroup$
    – IAmNoOne
    Commented Aug 23, 2016 at 14:30

3 Answers 3

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A graphical example of Robert Israel's explanation.

Some sets $A_1$, $A_2$, $A_3$, $A_4$ and their intersection $\cap_i^\infty A_i$.

Now we calculate their complementary (only the shaded parts).

Complementaries 1

Finally we join them and calculate the complementary of the resulting set.

Complementaries 2

We see that $\cap_i^\infty A_i = \big(\cup_i^\infty A_i^\complement\big)^\complement$

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    $\begingroup$ This is an amazing answer -- what software did you use to make these pictures/where did you find them? $\endgroup$ Commented Aug 23, 2016 at 15:10
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    $\begingroup$ Thank you. I've done it with CorelDRAW (it is propietary software). However, I suppose that it wouldn't be that difficult to do it with a free one such as Inkscape, for example. $\endgroup$
    – JnxF
    Commented Aug 23, 2016 at 15:13
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It's easier to visualize $4$ than $\infty$, but the principle is the same.

Draw $4$ circles (and make sure the intersection of all $4$ is nonempty). Note that this intersection is the only region that is not outside any of the circles.

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It’s easier, I think, to let $B_i=A_i^c$, so that your identity becomes $\bigcap_iB_i^c=\left(\bigcup_iB_i\right)^c$. Now draw your picture, with circles for the sets $B_i$. The set $\bigcap_iB_i^c$ consists of those points that are not in any of the circles, and so does the set $\left(\bigcup_iB_i\right)^c$.

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