TITLE: Can a set be a member of its own power set?
QUESTION [7 upvotes]: I'm new to set theory and I'm using actually an introduction to topology by Bert Mendelson that features some set theory questions. (side note, does anyone have a link to where the answers are for the books exercises?)
My question is the following, and give reasons:
true or false
For each set $A$, $A \in 2^A$

REPLY [2 votes]: Power set is nothing but the collection of all possible subsets of a given set. Clearly, $\emptyset$ and the set itself are included in the power set.
For example:
$A={1,2,3}$
Then the subsets of $A$ are:
{$\emptyset$}, {1}, {2}, {3}, {1,2}, {2,3}, {1,3} )and lastly, {1,2,3}
These are the elements in the power set of A represented as:
P(A) ={ {$\emptyset$}, {1}, {2}, {3}, {1,2},{2,3},{1,3},{1,2,3} }
Hope it helps!