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 [math]A^c = \{x \in \mathbb{R} \; : \; x \nin A\}[/math].

[math](A^c)^c = \{x \in \mathbb{R} \; : \; x \nin A^c\}[/math]. Now every x in A is in this set and every x in this set is in [math]A^c[/math]

 

2) Show that if [math]x \in (\bigcup_{i \in I} A_i)^c[/math] then it is in [math]\bigcap_{i \in I} A_i^c[/math] 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.