1. Prove DeMorgan's Laws. That is, let X be a set and let A and B be subsets of X. Show that (a) X – (AUB) = (X – A)n (X – B), and (b) X – (An B) = (X – A) U (X – B). a) X- CAUB) = (X - A)O(X- B) We have aE X- (A UB) So That is %3D ae X and a& A and agB. this only shows X – (AU В) сх — АПX - В. a¢ CAUB). QEX and 50, aE x-A and ae x-B. Hence ae Cx- A) N (x - B). Thus, X- LAU B) = \x-A)n(x-B) b) X- CANB) = (X - A)U (x- B) we have ae X- (ANBI so That is ae X and a& A or a¢B. a¢ CAn B). and this only shows X – (AN В) сх— АUX - B. 50, aE x-A a € x-B. Hence aG (x- A ) U(x-B). or Thus, X - LAN B) =(x-A)U(X-B)

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