I went to my home town during my vacations, and found that it was easier to lift a bucket of water with pulley as compared to that without pulley. Can anyone explain it mathematically assuming that pulley and rope is massless and frictionless?
The only thing that changes is the direction of the force needed. Without a pulley, you need to exert an upward force $F=mg$ on the bucket to lift it.
Using a ceiling mounted pulley you need to exert a downward force $F=mg$ on the rope to lift the bucket.
In both cases the magnitude of the force is the same.
But lifting or pulling down may subjectively be experienced differently by the person doing the lifting. Using a pulley it's also possible to pull the rope at an angle (to the horizon), which may also subjectively be experienced as 'easier'.