Proper subsets of $\{a,b,c,d\}$. List the members of $\mathcal P\left(\{a, b, c, d\}\right)$ which are proper subsets of $\{a, b, c, d\}$.
Sorry, I know this is basic, but I'm knew to this. I think the answer is just $\{a\}, \{b\}, \{c\}, \{d\}$. But im not too sure.
 A: There are $2^4$ elements (each a subset of $A = \{a, b, c, d\}$) in the power set of $\{a, b, c, d\}$ since the given set has $4$ elements. All but one of those subsets is proper. The power set of a set $A$ is simply the name we give to the set which contains all subsets of $A$.
A proper subset $B$ of a set $A$ is a subset $B\subseteq A$ with $B\neq A$.
Recall, for any set $A$, $A\subseteq A$ is a subset of itself, though this set is the one set that is NOT a proper subset. On the other hand, $\{a,b\}$ is a proper subset of $A = \{a,b,c,d\},$ as is $\varnothing \subseteq A$.
A: The power set $P(A)$ is defined to be the set whose elements are the subsets of $A$. So in this case, the power set will contain elements such as $\emptyset$, $\{a\}$, $\{b, c, d\}$, and so on. The proper subsets are just all the subsets which aren't the entire set. So the only non-proper subset is $\{a, b, c, d\}$.
As a hint to guide your answer, there are $16$ elements in the power set, in total.
A: You mean proper subsets? If so, a set $S$ is called a proper subset of another set $S'$ if $S\subseteq S'$, but there is some $s\in S'$ that is not in $S$. This is usually noted by $S\subsetneq S'$. For example, $$\{a,b,d\}$$ is a proper subset of your set. Indeed, any subset is proper when it is simply not the whole set (and if $S$ is nonempty, the definition implies the empty set is also a proper subset).
A: There are $2^{4}=16$ different subsets, $16-1=15$ of which are proper
subsets.
Note: A proper subset of a set $A$ is a subset $B\subset A$ but
$B\neq A$.
For example: $\{a,b\}$ is also a proper subset of $\{a,b,c,d\}$
