Prove independence preserved when replacing sets with complements

[kumamako]

Let A1,A2, . . . ,An be subsets of
. Show that if A1,A2, . . . ,An are independent, then the
same is true when any number of the sets Ai are replaced by their complements (Ai)c. (Hint:
First do the case in which just one of the sets is replaced by its complement. Then argue by
induction on the number of sets replaced.)

Can someone guide me through this question please?

thanks

 

[HallsofIvy]

What is your definition of "independent" sets?