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Question

Prove that the number of subsets of a set containing n distinct elements is 2n for all nϵN.

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Solution

Let P(n) : Number of subset fo a set containing n distinct elements is 2n, for all nϵN.

Step 1 we observe that P(1) is true, for n = 1

Number of subsets of a set contain 1 element is 21=2, which is true.

Step II Assume that P(n) is true for n = k

P(k) : Number of subsets of a set containing k distinct elements is 2k, which is true.

Step III To prove P(k + 1) is true, we have to show that

P(k + 1) : Number of subsets of a set containing (k + 1) distinct elements is 2k+1. We know that, with the addition of one element in the set, the number of subsets become double.

Number of subsets of a set containing (i + 1) distinct elements =2×2k=2k+1. So, P(k + 1) is true. Hence, P(n) is true.


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