How to compare dates in datetime fields in Postgresql?

I have been facing a strange scenario when comparing dates in postgresql(version 9.2.4 in windows).

I have a column in my table say update_date with type timestamp without timezone.
Client can search over this field with only date (e.g: 2013-05-03) or date with time (e.g.: 2013-05-03 12:20:00).

This column has the value as timestamp for all rows currently and have the same date part 2013-05-03, but difference in time part.

When I’m comparing over this column, I’m getting different results. Like the followings:

select * from table where update_date >= '2013-05-03' AND update_date <= '2013-05-03' -> No results

select * from table where update_date >= '2013-05-03' AND update_date < '2013-05-03' -> No results

select * from table where update_date >= '2013-05-03' AND update_date <= '2013-05-04' -> results found

select * from table where update_date >= '2013-05-03' -> results found

My question is how can I make the first query possible to get results, I mean why the 3rd query is working but not the first one?