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I am using only free, open source tools. I have two shapefiles: a shapefile of voting precincts and a shapefile of census tracts. I want to know: which voting precincts fit inside which census tracts? What is the mapping vocabulary describing what I am trying to answer?

I am trying to know which polygons from the precinct shapefile are bounded by which polygons from the census shapefile? Is that right? How do I answer that question programmatically? What tools would I use? I have a vague sense that I could do this in PostGIS -- but don't really know. Also, is it possible in QGIS?

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You can achieve this using PostGIS spatial functions.

St_Within will return True if geom A is totally within geom B. There are other similar functions you can use if this isn’t suitable, check the docs.

SELECT census_tracts.col1, voting_precincts.col1
FROM census_tracts, voting_precincts
WHERE st_within(census_tracts.geom, voting_precincts.geom);
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What you are looking for is ST_Contains and ST_Within functions. Your use case is pretty basic, so I recommend reading through the docs. Once you bump into a problem you can't solve yourself even after reading it, ask here.

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