1.

N(A)=n then find the number of elements in power set A​

Answer»

Answer:

If a non-empty SET A contains n elements, then its power set contains 2  

n

 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=2  

0

.

Now, suppose ∣A∣=n.

By induction hypothesis, we know that ∣P(A)∣=2  

n

⟶1

Let 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 2  

n

 of them.

The second one are of the form  C∪{a}, where C∈P(A).  since there are 2  

n

 possible choices for C, there must be exactly 2  

n

 subsets of B of which    

′

a  

′

  is an element.

∴∣P(B)∣=2  

n

+2  

n

=2  

n+1

.

so, if set has n elements, then power set has  2  

n

  elements.

Hence proved.

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