Induction of Complementary Sets

[rbzima]

I'm just wondering how induction can be used to show the following:

(A_1 \bigcup A_2 \bigcup A_3 \bigcup \cdots \bigcup A_n)^c = A^c_1 \bigcap A^c_2 \bigcap A^c_3 \bigcap \cdots \bigcap A^c_n

 

[Werg22]

What's the base case? Did you try to establish whether a case follows if the previous is true (if the complement of the union of k sets is equivalent to the intersection of their respective complements, does this imply the same for k + 1 sets?).

 

[mathman]

Once you show it for two sets, it is easy. n+1 sets can be considered as 1 set union with n sets. Complement this gives then the complement of 1 set intersected with the complement of the union of n sets.