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Question
Prove that number of subsets of a set containing n distinct elements is 2n, for all n ∈ N.
Solution
Let P(n) : Number of subsets of a set containing n distinct elements is 2n, ∀ n ∈ N
Step 1: It is clear that P(1) is true for n = 1.
Number of subsets = 21 = 2.
Which is true.
Step 2: P(k) is assumed to be true for n = k.
Since the number of subsets = 2k.
Step 3: P(k + 1) = 2k + 1
We know that if one number (i.e., element) is added to the elements of a given set, the number of subsets become double.
∴ Number of subsets of set having (k + 1) distinct elements = 2 × 2k = 2k + 1
Which is true for P(k + 1).
Hence P(k + 1) is true whenever P(k) is true.
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