Let S be a set with n elements. Show:

If an (n is subscript) equals the number of subsets of S then an+1 (n+1 is subscript) = 2an (n is subscript)

Use this to prove by induction that an (n is subscript) = 2^n

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- Jun 20th 2010, 12:56 PMhuntstyClueless on this proof, please help!!!
Let S be a set with n elements. Show:

If an (n is subscript) equals the number of subsets of S then an+1 (n+1 is subscript) = 2an (n is subscript)

Use this to prove by induction that an (n is subscript) = 2^n - Jun 20th 2010, 01:24 PMPlato
- Jun 20th 2010, 01:53 PMhuntsty
thank you, very helpful for understanding. I just don't know how to symbolically show this. Will i need to introduce a new random variable in my proof?

- Jun 20th 2010, 02:03 PMPlato
- Jun 21st 2010, 12:02 PMHallsofIvy
If S contains n+ 1 members, choose one of them and call it "a". Removing that from S leaves you with set T that has n members and so, a(n) members.

Now note that every member of S either contains "a" or it doesn't. If it doesn't, it is one of the a(n) subsets of T. If it does, then it is one of the subsets of T with "a" added- there are still a(n) such subsets. Together there are a(n)+ a(n)= 2a(n) subsets of S.