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**Possible actuary** Given: Show that for any integer n greater than or equal to 2 given n sets A1,A2,.....An then the complement (A1 U A2 U...An) = complement A1 & complement A2 & ... complement An.

Proof: By DeMorgan's Law, complement (A1 U A2) = complement A1 & complement A2, so the result is true for n=2. Now suppose that for k greater than or equal to z, then the complement of the union of any k sets is the intersection of the complements. That is if A1, ... Ak are sets, then complement (A1 U ... Ak) = complement A1 & ... complement Ak. Let S1,S2,...Sk, S(k+1), be complement (S1 U S2 U ...Sk U S(k+1)) = complement (S1 U S2 U ... Sk & complement S(k+1) {by DeMorgan's Law} = complement S1 & complement S2 & ... complement Sk & complement S(k+1) {by the induction hypothesis}. Thus th result is true for k+1.

A1, S(k+1), An the 1, n, and (k+1) are subscripts.

Can this proof be done by not including sets of S? If not, then what does stating k is greater than or equal to z mean?