1. ## Powerset theory

This is kind of a simple question. Anyways, I have a set S = {1, 2, 3, 4}

I know the powerset has 2^4 elements which is 16. I'm not sure how to find out how many of the sets in the powerset contain a certain element. I know the answer is 8 for any element.

For example, how many of the sets in the powerset have a 1 in it. I just wrote them all out and counted and the answer is 8:
{}, {1}, { 2}, {1, 2}, { 3}, {1, 3}, { 2, 3}, {1, 2, 3}, { 4}, {1, 4}, { 2, 4}, {1, 2, 4}, { 3, 4}, {1, 3, 4}, { 2, 3, 4}, {1, 2, 3, 4}

...but how do I calculate this without writing them all out?

edit: Well I wrote up some different sets, and it always seems to be half of the number of total subsets. Is there a reason why it does this?

Thanks.

2. Originally Posted by swtdelicaterose
This is kind of a simple question. Anyways, I have a set S = {1, 2, 3, 4}

I know the powerset has 2^4 elements which is 16. I'm not sure how to find out how many of the sets in the powerset contain a certain element. I know the answer is 8 for any element.

For example, how many of the sets in the powerset have a 1 in it. I just wrote them all out and counted and the answer is 8:
{}, {1}, { 2}, {1, 2}, { 3}, {1, 3}, { 2, 3}, {1, 2, 3}, { 4}, {1, 4}, { 2, 4}, {1, 2, 4}, { 3, 4}, {1, 3, 4}, { 2, 3, 4}, {1, 2, 3, 4}

...but how do I calculate this without writing them all out?

edit: Well I wrote up some different sets, and it always seems to be half of the number of total subsets. Is there a reason why it does this?

Thanks.
If you have elements $\{1,\cdots,n\}$ with $2^n$ subsets then the number of subsets you an form without, say $n$, is the number of subsets you can form from $\{1,\cdots,n-1\}$ which is $2^{n-1}$.

$\frac{2^{n-1}}{2^{n}
}=\frac{1}{2}$