Why does the moon and the sun appear in the day sky and the moon is not full?

The Moon rises about 50 minutes later each night. After a few weeks, it is logically rising in the morning, and crossing the sky during the day. As the month goes on, it eventually is rising at night again. The moons movements are independent of the sun. Though day length varies, the sun is always up during the "day", and gone at night. Not so the moon.