# Prove $\overline{\overline{A}\cap{\overline{B}}}\cup\overline{\overline{A}\cap{B}}=A$

How do you prove this proposition from set theory, where $$\overline X$$ denotes the complement of $$X$$?

$$\overline{\overline{A}\cap{\overline{B}}}\cup\overline{\overline{A}\cap{B}}=A$$

My gut feeling tells me to apply De Morgan Law but I don't know how to go about this. Thank you!

• Does the overline denote complement or closure? – mxnoqwerty Oct 8 '18 at 0:13
• @mrnoqwerty - I rather suspect complement – Henry Oct 8 '18 at 0:14
• It denotes complement. – James Warthington Oct 8 '18 at 0:15
• This is not true. The left-hand side is the entire universe. – David Peterson Oct 8 '18 at 0:25

This is false if your space is $$\{0\}$$ and $$A=B=\emptyset$$. Because $$\overline{A}=\overline{B}=\{0\}$$, the left hand side becomes $$\overline{\{0\}}\cup\overline{\emptyset}=\emptyset\cup\{0\}=\{0\}$$ which is not $$A$$.
What is true, is that $$\overline{\overline{A}\cup\overline{B}}\cup\overline{\overline{A}\cup B}=A$$.
If we have $$\overline{V\cup W}$$, this is everything not in $$V\cup W$$, so everything neither in $$V$$ nor $$W$$. Hence $$\overline{V\cup W}=\overline{V}\cap\overline{W}$$. So we get $$\overline{\overline{A}\cup\overline{B}}\cup\overline{\overline{A}\cup B}=(A\cap B)\cup(A\cap\overline{B})=A\cap(B\cup\overline{B})=A$$.