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This seems like it should be fairly simple, but have been having quite a bit of trouble. I am no expert in ArcGIS for Desktop

I have: one layer of points, and 4 polygons, which partly overlap

I can select points based on a polygon, however what I want to do is:

I would like to add 4 columns to the points attributes table (to export), with 1s and 0s of whether the point falls inside or outside each of the polygons separately. Is that possible?

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You can add a column to the attribute table of the polygons and have it contain a 1 for all records in the table. Then, split the polygon layer to four individual shape files. Next, conduct a spatial join between your points layer and these four new layers with the parameter for the join set to "within". Its a bit roundabout but all the points falling within the polygons should have a "1" listed in a separate column for each polygon. That is how I would attempt a project like this on QGIS, which is what I currently use, but in the past I have used ESRI and don't see that there should be much difference in process between the two.

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