De Morgan's laws for events
/ duh-MOR-gunz /
De Morgan's laws are two simple rules that tell you how the complement (the 'not') interacts with unions ('or') and intersections ('and'). In words: the event that 'not at least one of A, B happens' is the same as 'neither A nor B happens', and the event that 'not both A and B happen' is the same as 'A fails or B fails'. They are the formal version of an intuition you already use when you negate an English sentence about chances.
In symbols the two laws read (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c. Notice the pattern: pushing a complement inside a parenthesis flips union into intersection and intersection into union, and complements every piece. A picture confirms it — the outside of the combined region 'A or B' is exactly the place that is outside A and outside B at once. The laws extend to any number of events: the complement of a big union is the intersection of all the complements, and vice versa.
These laws are the everyday tool of complementary counting, which is the single most useful trick in elementary probability. The phrase 'at least one' is awkward to compute head-on, but its negation is the clean event 'none', and De Morgan is what licenses the swap: P(at least one of the A's) = 1 - P(none of them) = 1 - P(all the complements happen together). Whenever a problem says 'at least one', reach for the complement and let De Morgan rewrite it for you.
Roll two dice. To find P(at least one six), rewrite the event with De Morgan: 'at least one six' is the complement of 'no six on either die'. P(no six) = (5/6)(5/6) = 25/36, so P(at least one six) = 1 - 25/36 = 11/36.
'At least one' is hard head-on but easy as 1 minus 'none' — De Morgan is what justifies that flip.
The flip is the whole point: complement of a union is the intersection of complements, never the union of complements — mixing them up is the common slip.