Question on Set complement

[bloodhound]

Hi ya. can anyone give me a hint on how to start showing the following

:For any artitary subsets of R

1)a complement of a complement is the original set
2)complement of a union is the intersection of the complements.

cheers.

[MandrakeRoot]

Just show that the two sets are equal.

1) Let A be any subset of R, then A^c = \{x \in \mathbb{R} \; : \; x \nin A\}.
(A^c)^c = \{x \in \mathbb{R} \; : \; x \nin A^c\}. Now every x in A is in this set and every x in this set is in A^c

2) Show that if x \in (\bigcup_{i \in I} A_i)^c then it is in \bigcap_{i \in I} A_i^c and vice versa.

The first part would be an argument of the type, that if x is in the complement of your union, then it is not in the union, hence in none of the A_i, therefore it would be in all of the complement of A_i and thus also in the intersection. The other way around goes along the same lines.

Mandrake

[pulkit]

The second part can be done using induction as well, for its easier to prove for 2 sets and then do inductively.