How should a brother deal with his adulterous sister? Should he leave her to continue committing this sin or should he expel her from the house and end his relationship with her?

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.

Enjoining what is good and forbidding what is evil is a duty for anyone who can do it, if no one else did. The Prophet, sallallaahu ‘alayhi wa sallam, ordered the one who sees an evil to change it physically with his hand if he was able to do so. If he could not, he should change it by his tongue (by advising and/or condemning), and finally by (detesting it with) his heart, and he, sallallaahu ‘alayhi wa sallam, informed that this is the weakest level of faith. [Muslim]

This woman has committed a major grave sin and her brother should not approve of this sin, and should prevent her from committing this sin by keeping her at home and preventing her from leaving (if he is able to do so). Moreover, he should advise her and remind her of the punishment of Allaah The Almighty. It is not permissible to allow her to act as she wishes, and it is also not permissible to throw her out of the house as this would only spread corruption.

Allaah Knows best.