Here is my question.
There exist a finite set A that equals its own Power Set ?
True or False
If Power Set has any elements,
since Power Set definitely has empty set ,
There never exist any finite set which equals its own Power Set.
The only I have considered is that empty set.
but since P(∅)={∅, {∅}} so above states is still false. right?
Please if I am doing something wrong, please advice me. Thanks
Best Answer
False!
This answer will be written for someone just beginning their explorations of set theory. If this is not you: feel free to peruse the other answers.
Welcome to the exploration of sets! You're in for a wild ride if you stick with it...
Proof
Left to the asker... but here are some hopefully helpful hints.
As is noted in the comments: $P(\emptyset) =\{ \emptyset\}$
Note that $\emptyset=\{\}$ has cardinality zero: It has no elements. Whereas $P(\emptyset)$ has cardinality 1. That is, $P(\emptyset)$ is a set with one element in it. That one element is $\emptyset$.
Symbolically we may write $0=|\emptyset|\neq |P(\emptyset)|=1$ where $|S|$ is the number of elements in a set. We can argue that $\emptyset \neq P(\emptyset)$ because they have different cardinalities.
Foreshadowing: We may continue (ad infinitum) arguing that $A \neq P(A)$ because $|A| \neq |P(A)|$ for $ A \neq \emptyset$.
Taking $A= \{1,2,3 \}$ you should find that $3=|A|\neq |P(A)|=8$. This can be seen here. For any finite set you should be able to find a formula for $|P(A)|$. I leave it to you to try this for all finite sets $A$. Yes! All of them! You can do this for all finite sets! It really shouldn't take you to long to find out the size of of every single finite powerset. You start by making an equivalence class so you don't have to deal with $\{1,2,3\}$ and $\{red,blue,green\}$ as separate cases.
Then look at some small $n$ to derive $|P(\{ 1,2,3 \dots, n \})|$.
Good luck. Have fun.