De Morgan's Laws

De Morgan's Laws describe how the complement of a union or intersection relates to the complements of the individual sets — they are among the most useful identities in this chapter, and come up constantly in later probability and logic contexts as well.

In words: "the complement of a union is the intersection of the complements," and "the complement of an intersection is the union of the complements" — notice the operation flips (union becomes intersection, and vice versa) each time a complement is taken across it.

The figure above places every element of Worked Example 7's sets — $U=\{1,\dots,10\}$, $A=\{1,2,3,4\}$, $B=\{3,4,5,6\}$ — into its correct region: $1, 2$ inside $A$ only, $5, 6$ inside $B$ only, and $3, 4$ shared by both. Everything outside both circles — $7, 8, 9, 10$, never drawn inside either circle — is exactly $(A \cup B)'$, and it is also exactly $A' \cap B'$: the same region reached two different ways, which is what De Morgan's Law asserts. …