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Power Set

Illustration
Let A = {1, 2, 3}. Let us write all the subsets of A. The subsets of A are φ, {1}, {2}, {3}, {2, 3}, {1, 3}, {1, 2} and {1, 2, 3}. The set of all subsets of A is called as power set of A. It is denoted by P(A). In the present case, P(A) = {φ, [1}, {2}, {3}, {2, 3}, {1, 3}, {1, 2}, {1, 2, 3}}.  

Here in this illustration n(A) = 3, then n [ P (A) ] = 8 = 23

In general, if A is a set with n(A) = m, then it can be shown that n[ P(A)] = 2m.

Note that A is not a subset of B if there is at least one element in A which is not an element of B. Furthermore, we write A B if A is not a subset of B. 

Illustration
Let A = {a, e, i, o, u}, and B = {a, b, c, d, f}. Note that A B, since e A but e B. Also, note that B A, since d B but d A.




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