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sql - 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 (i.e: 2013-05-03) or date with time (i.e: 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?

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@Nicolai is correct about casting and why the condition is false for any data. i guess you prefer the first form because you want to avoid date manipulation on the input string, correct? you don't need to be afraid:

SELECT *
FROM table
WHERE update_date >= '2013-05-03'::date
AND update_date < ('2013-05-03'::date + '1 day'::interval);

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