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I have several circles that are set at type none color and 0.01 black stroke. They're all currently in paths (except two I made into compound paths). I want to know how to remove the stroke from the areas where the shapes overlap.
Here is an image:

I want it to look like what's on the right.

What I've done so far: I can get somewhere if I select the two hollow circles which are currently 4 separate paths and make them into a compound path on exclude, but it still looks like the image on the left. I tried removing the anchors, but that does something completely different since those two circles are then broken up into a total of 7 pieces at the intersection points.

19

If the lower arcs are closed you can Select all the shapes, go to Pathfinder (Window -> Pathfinder) and choose Unite function

Two upper circles are compound path

enter image description here

enter image description here

enter image description here

21

Select the shapes and click-drag with the Shape Builder Tool to easily combine areas.

shape builder

Note: That last click-drag I performed isn't really needed :)

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    I deleted my answer, cause your is 46k times better – Ilan Apr 9 '14 at 20:37
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    @Ilan multiple, different answers are encouraged. You never know what answer a user will find most beneficial. I'd suggest you undelete it ;) – Scott Apr 9 '14 at 20:38
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    @Scott: completely off topic but... what do you use to make this awesome screen grab GIFs? – cockypup Apr 9 '14 at 21:45
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    I was going to ask about the GIF thing too! I really like graphic based tutorials like what you and Ilan had posted. Everyone does video tutorials now. Anyway, what are good sites to look into for illustrator that goes into the same approach the both of you did to answer this question? I think both of your answers are great. – user2899444 Apr 9 '14 at 23:29

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