(a) Let S be a set, and define the set W as follows: Basis: ∅ ϵ W.Recursive Definition: If x ϵ S and A ϵ W, then {x} U A ϵ W. Provide an explicit description of W, and justify your answer.
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Let us show that .
Let us prove using mathematical induction on the order of the elements of .
Base case:
n=0: is a unique subset of of order 0, and by defenition of , .
n =1: for each the singleton , and thus contains all singletons.
Inductive step:
Assume that contains all subsets of cardinality , and prove that it is also contains all subsets od cardinality . Indeed, let arbitrary subsets of cardinality . Then by assumption, and consequently, .
Conclusion:
Therefore, by Mathematical Induction .