If A-union-B = A'-union-B', prove that A = B.

[Enh0702]

Prove that if A and B are sets such that (A union B) ' = A' union B' then A=B

' this means the complement

This has been giving me a lot of trouble

I started by saying suppose x is an element of (A union B)'. Then since x is an element of that x is an element of A' union B' since they are equal. Ten x is an element of A' and x is an element of B'. Since A' intersect B' = A' union B' then A'=B'.

This is what I've done but I don't know if Im on the right track or not? Any help is appreciated.

 

[pka]

Given \(\displaystyle \left( {A \cup B} \right)^\prime = A' \cup B'\).
If \(\displaystyle x \in A\) then
\(\displaystyle \begin{array}{rcl}
x \in A \cup B & \Rightarrow & \quad x \notin \left( {A \cup B} \right)^\prime \\
& \Rightarrow & \quad x \notin \left( {A' \cup B'} \right) \\
& \Rightarrow & \quad x \notin A' \wedge x \notin B' \\
& \Rightarrow & \quad x \in B \\
\end{array}\).
That shows that \(\displaystyle A \subseteq B\)!

Can you show that \(\displaystyle B \subseteq A\)?
What does all this prove ?