Q:

Is There a Way to Search by the Opposite of a Category in Magento? 

I am wondering if there is a way a magento developer would be able to create or find something that would search by the opposite of a category.   
For example, given 100 total products and a category x which holds 70 products, is there a way to know what the remaining thirty products are?   Besides, of course, crossing off category x's products one by one on a list of the total number of products. 

A:

You should be able to query the DB.  Filtering on the category_id that you want.  I don't think there will be a better way. 
See: catalog_category_product and note that there are columns: category_id and product_id. 
You should be able to use the category_product resourceModel to set this up.  Set the filter to be ! = to the category_id you don't want. 
If you have trouble with the actual code, open a new question.