A is a set containing n elements. A subset P 1 is chosen, and A is reconstructed by replacing the elements of P 1 . The same process is repeated for subsets P 1 , P 2 , … , P m , with m > 1. The Number of ways of choosing P 1 , P 2 , …, P m so that P 1 ∪ P 2 ∪ … ∪ P m = A is -
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Let A = {a 1 , a 2 ,…..a n }
For each a i (1 ≤ i ≤ n), either a i ∈ P j or a i ∉ P j (1 ≤ j ≤ m) . Thus, there are 2 m choices in which a i (1 ≤ j ≤ n) may belong to the P j ′ s.
Also there is exactly one choice, viz., a i ∉ P j for j = 1, 2, …, m, for which a i ∉ P 1 ∪ P 2 ∪ ... ∪ P m .
Therefore, a i ∈ P 1 ∪ P 2 ∪ …. ∪ P m in (2 m – 1) ways . Since there are n elements in the set A, the number of ways of constructing subsets
P 1 , P 2 , ….. , P m is (2 m – 1) n
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