Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
Is it permissible for a young man to marry a young woman who is only seven months older than him?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
From an Islamic perspective, it is permissible for a man to marry an older woman. And Allah The Almighty Knows Best..
Is it permissible for a woman to visit the doctor while in her `Iddah (waiting period)?
It is permissible for her to do that during day time, but it is impermissible for her to leave her house at night except for a sound reason. And Allah Knows Best.
Is it permissible for a woman who has become pure before the end of forty days of confinement to perform different acts of worship? and is it lawful for her husband to have sex with her?
Once postpartum bleeding ceases and she is certain that it won`t happen again, she becomes pure, and so she should perform Ghusl (ritual bath), pray, and fast. She can also have sex with her husband even before the end of (40) days because the minimum of postpartum bleeding is a moment, and the maximum is (60) days. However, the postpartum bleeding for the majority of women is (40) days, but this does not apply to rest of them.
