Homework Help: A proof in Set theory about the number of different subset in a Set

hi guys, I am a physics major, recently doing a few self-reading on analysis, however, I got stuck at some excises of proofs.

1. The problem statement, all variables and given/known data
If a set A contains n elements, prove that the number of different subsets of A is equal to [itex]2^n[/itex].

3. The attempt at a solution

First, I teared apart the problem, attempting to draw a general equation from looking at the numbers of different subset one by one.

E(m) regarded as the possible number of different subsets which contains m elements, whereas A has total n elements.

[tex]E(0)=1[/tex]
[tex]E(1)=n[/tex]
[tex]E(2)=\frac{n^2}{2}-\frac{n}{2}[/tex] (this is obtained from a bit of analysis. Possibly be fault however.)
[tex]E(3)=\frac{5n^3}{6} -\frac{n^2}{2}-\frac{n}{3}[/tex] (so is it.)
.
.
.
[tex]E(n)=1[/tex]

However, I was almost driven crazy... I could't make the generl equation from it (which is hopefully [itex]2^n[/itex])

Any other approaches? Or any mistakes I made during my approach?

Thanks for reading!

 

Thanks for answering!
However, I would love to know how to prove nCm suitable here?

 

Probably my knowledge in Math induction is not enough.
I have attempted to prove it via M.I.
however, I was having some hard time

my approach:
1.) obviously it is true for n=1,0
2.) I assume that it is true for [itex]2^k[/itex], where k belongs to natural set.
3.) but how do I know/prove it is also true for [itex]2^{k+1}[/itex]? since I can't write such a function at all!

 

besides, I always do not understand how the nCm stuffs have to do with the so-called "possibilities" of combinations?
thanks!