Merging separate paths in Illustrator


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.

4/10/2014 5:16:00 AM

Accepted Answer

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

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4/9/2014 8:57:00 PM

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 :)