Thread: Power sets

I have a homework problem that I knew the answer but dont know how to do the proof. Can anyone help me?

List all the subsets of the set X={a,b,c}. How many are there? Next, try with X={a,b,c,d). Now suppose that the set X has n elements: how many subsets does X have? Try to justify your answer.

I do the first part as
For X={a,b,c}
{},
{a},{b},{c},
{a,b},{a,c},{b,c}
{a,b,c}
which is 1,3,3,1
For X={a,b,c,d}, in the same pattern, which is 1,4,6,4,1.

I think it is the Pascal triangle's pattern and the answer is 2^n, but I have no idea how to write the proof.

Originally Posted by r7iris

I have a homework problem that I knew the answer but dont know how to do the proof. Can anyone help me?

List all the subsets of the set X={a,b,c}. How many are there? Next, try with X={a,b,c,d). Now suppose that the set X has n elements: how many subsets does X have? Try to justify your answer.

I do the first part as
For X={a,b,c}
{},
{a},{b},{c},
{a,b},{a,c},{b,c}
{a,b,c}
which is 1,3,3,1
For X={a,b,c,d}, in the same pattern, which is 1,4,6,4,1.

I think it is the Pascal triangle's pattern and the answer is 2^n, but I have no idea how to write the proof.

It is 2^n.

Consider,


There is the one empty set, which is

There are singleton sets, which is

There are two elements sets which is .

And so on ...

Until we use all the elements, and the full set itself which is .

In total we have,


See if you can figure out the last line.