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## Homework Statement

If a set A contains n elements, prove that the number of different subsets of A is equal to [itex]2^n[/itex].

## 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.)

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[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!