The complement of <em>A</em> ∪ <em>B</em> is the set of all elements not in <em>A</em> ∪ <em>B</em>.
If <em>x</em> is some element of <em>A</em> ∪ <em>B</em>, then <em>x</em> is either in <em>A</em> or <em>B</em> (or both).
So if <em>x</em> is some element of the complement of <em>A</em> ∪ <em>B</em>, that means <em>x</em> does not belong to <em>A</em> or <em>B</em>. In other words, <em>x</em> is not in <em>A</em> and <em>x</em> is not in <em>B</em>.
So (<em>A</em> ∪ <em>B</em>)<em>'</em> = <em>A'</em> ∩ <em>B'</em>, where the <em>'</em> symbol denotes set complement.
For example, if <em>A</em> = {1, 2, 3, 4} and <em>B</em> = {3, 4, 5, 6} are both subsets of <em>U</em> = {0, 1, 2, 3, 4, 5, 6, 7}, then
<em>A</em> ∪ <em>B</em> = {1, 2, 3, 4, 5, 6}
→ (<em>A</em> ∪ <em>B</em>)<em>'</em> = {0, 7}
Put another way,
<em>A'</em> = {0, 5, 6, 7}
<em>B'</em> = {0, 1, 2, 7}
→ <em>A'</em> ∩ <em>B'</em> = {0, 7}
and we see that both (<em>A</em> ∪ <em>B</em>)<em>'</em> and <em>A'</em> ∩ <em>B'</em> are the same set.