Why Does the Power Set of a Set Have 2^n Elements?

[michonamona]

Homework Statement

Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations

The Attempt at a Solution

 

[Robert1986]

Homework Statement

Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations

The Attempt at a Solution

Haha, keep doing combinatorics and a lot of stuff will blow your mind.


I don't want to give too much away, here, I'll be around to help if you need it, but think about the set and the power set of that set as a binary string of length n where each element of the string represents an element of the set.

 

[LCKurtz]

Homework Statement

Why is the size of the power set 2^n ?

To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ?

It boggles my mind why the base is 2 for all size of sets.

Thank you,

M

Homework Equations

The Attempt at a Solution

If you have a set with n elements, now many subsets of size 0 are there? Of size 1? Size 2?...Size n? How many total then?

Then think about the binomial expansion of (1+1)ⁿ.