List all subsets of {a, b, c, d, e}, containing a but not containing b I wonder...can I solve this by just getting A = {a, c, d, e} minus those subsets which do not contain a?
So, let A be {a, c, d, e}. |A| is 2^4 = 16
And let B be {c, d, e}. |B| is 8
And let C be subset of subsets of {a, b, c, d, e}, containing a but not containing b
Can I conclude that that |C| will be 8, that is, |A| - |B|?
I could enumerate these:


*

*{a}

*{a, c}

*{a, d}

*{a, e}

*{a, c, d}

*{a,c, e}

*{a, d, c}

*{a, d, e}


Which are 8, not including the empty set so I guess there is some mistake with my reasoning. Could you people point that out for me?
P.S: this question is from the book Discrete Mathematics: Elementary and Beyond, by Lovász et al.
 A: All you need to do is list all subsets of $\{c,d,e\}$, and then add $a$ to each of those.  There are eight of them.  (You've listed $\{a,c,d\}$ twice though, and you missed one of the eight.)
A: You missed $\{a,c,d,e\}$. And, you have listed $\{a,c,d\}$ twice. This is because in set notation, the order of the elements within the brackets does not matter (so $\{a,c,d\}$ is the same set as $\{a,d,c\})$
A: An important concept is something called a "power set". If $S$ is a set, then the power set of $S$ is the set of all subsets of $S$, and may be written $P(S)$, $\mathcal P(S)$, or $\mathscr P(S)$, depending on taste.
For example, $$\mathcal P(\varnothing)=\{\varnothing\}$$ and $$\mathcal P(\{1,2\})=\{\varnothing,\{1\},\{2\},\{1,2\}\}.$$
List out $|\mathcal P(S)|$ for $S=\{1,\ldots,n\}$ for a few small values of $n$.
Now think about how you might represent the power set of $S$ as a finite sequence of $1$s and $0$s, that is, as a bit vector.
Can you see why the power set has the size it does for each $n$?
Continuing your education …
If $S$ and $T$ are sets, then $S^T$ is the set of all functions from $T$ into $S$.
Mathematicians often like to define $2:=\{ 0, 1 \}$.
Can you see how $2^T$ relates to $\mathcal P(T)$? Can you see how $|2^T|$ relates to $|T|$?
A: An important concept is something called a "power set". If SS is a set, then the power set of SS is the set of all subsets of SS, and may be written P(S)P(S), P(S)P(S), or P(S)P(S), depending on taste.
For example,
P(∅)={∅}
P(∅)={∅}
and
P({1,2})={∅,{1},{2},{1,2}}.
P({1,2})={∅,{1},{2},{1,2}}.
List out |P(S)||P(S)| for S={1,…,n}S={1,…,n} for a few small values of nn.
Now think about how you might represent the power set of SS as a finite sequence of 11s and 00s, that is, as a bit vector.
Can you see why the power set has the size it does for each nn?
