Prove by induction onn: If a setAhasnelements, its power set has 2^nelements.

*Given a setA, thepower setofAis the set of all subsets ofA

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- Nov 27th 2010, 05:38 PMthamathkid1729Cantor's Theorem
Prove by induction on

*n*: If a set**A**has*n*elements, its power set has 2^*n*elements.

*Given a set**A**, the*power set*of**A**is the set of all subsets of**A** - Nov 27th 2010, 05:51 PMArchie Meade

**P(k)**

The power of a set with k elements is

**P(k+1)**

The power of a set with k+1 elements is

Show that P(k+1) must be true if P(k) is true.

If a set of k elements has a power of

then an added element will form new subsets as it can be placed with all existing subsets.

Therefore a set of k+1 elements has a power of if a set of k elements has power

Finally prove for the base case. - Nov 27th 2010, 06:47 PMPlato
This proof turns on a simple fact:

Any subset of is a subset of

So by uniting any subset of with we get a subset of .