    Question

# If a non-empty set A contains n elements, then its power set contains how many elements?

A
n2
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B
2n
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C
2n
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D
n+1
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Solution

## The correct option is D 2nIf a non-empty set A contains n elements, then its power set contains 2n elements.This can be proved using mathematical induction.Base Case: suppose |A|=0⟹A=ϕ. But, empty set is only subset of itself. So, |P(A)|=1=20.Now, suppose |A|=n.By induction hypothesis, we know that |P(A)|=2n⟶1Let B be a set with (n+1) elements, B=A∪{a}Now, there are 2 kinds of subsets of B: those that include ′a′ and those that don't.The first ones are exactly the subsets of X which do not contain ′a′ and there are 2n of them.The second one are of the form C∪{a}, where C∈P(A). since there are 2n possible choices for C, there must be exactly 2n subsets of B of which ′a′ is an element.∴|P(B)|=2n+2n=2n+1.so, if set has n elements, then power set has 2n elements.Hence proved.  Suggest Corrections  0      Similar questions  Related Videos   Power Set
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