PostgreSQL supports the SQL OVERLAPS operator. You can use it to test if two date ranges overlap:

=> SELECT ('2001-02-16'::date, '2001-12-21'::date) OVERLAPS
          ('2001-12-20'::date, '2002-10-30'::date);

overlaps
--------
true

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An important caveat is that the date ranges are defined as start <= time < end. As such the later date is not included in the range:

=> SELECT ('2001-02-16'::date, '2001-12-21'::date) OVERLAPS
          ('2001-12-21'::date, '2002-10-30'::date);

overlaps
--------
false

Also compare our card Test if two date ranges overlap in Ruby or Rails.