How do De Morgan’s laws relate complements to unions and intersections?

"• De Morgan’s laws. Now these are worth memorizing, if only because (1) they’re incredibly important, and (2) they may not slip right off the tongue the way the previous properties do. The first one can be stated this way: X ∪ Y = X ∩ Y . Again, it’s best understood with a specific example. Let’s say you’re renting a house, and want to make sure you don’t have any surly characters under the roof. Let X be the set of all known thieves. Let Y be the set of all known murderers. Now as a landlord, you don’t want any thieves or murderers renting your property. So who are you willing to rent to? Answer: if Ω is the set of all people, you are willing to rent to X ∪ Y . Why that? Because if you take X ∪ Y , that gives you all the undesirables: people who are either murderers or thieves (or both). You don’t want to rent to any of them. In fact, you want to rent to the complement of that set; namely, “anybody else.” Putting an overbar on that expression gives you all the non-thieves and non-murderers. Very well. But now look at the right hand side of the equation. X gives you the non-thieves. Y gives you the non-murderers. Now in order to get acceptable people, you want to rent only to someone who’s in both groups. Put another way, they have to be both a non-thief and a non-murderer in order for you to rent to them. Therefore, they must be in the intersection of the non-thief group and the non-murderer group. Therefore, the two sides of this equation are the same. The other form of De Morgan’s law is stated by flipping the intersections and unions: X ∩ Y = X ∪ Y . Work this one out for yourself using a similar example, and convince yourself it’s always true. Augustus De Morgan, by the way, was a brilliant 19 th century mathematician with a wide range of interests. His name will come up again when we study logic and mathematical induction."