# Thread: Another set theory

1. ## Another set theory

Let X be a finite set. Show that P(X) has 2^|x| elements. Also, show that |2^x|=2^|x|.

{Hint: to give a rigorous proof, induct on the number of elements of X.}

2. Originally Posted by r7iris
Let X be a finite set. Show that P(X) has 2^|x| elements. Also, show that |2^x|=2^|x|.

{Hint: to give a rigorous proof, induct on the number of elements of X.}
what does small x represent? and by P(X) you mean the power set of X correct?

3. Originally Posted by Jhevon
what does small x represent? and by P(X) you mean the power set of X correct?
|x| is the number of member in set X.
P(X) is the power set of X.

4. A subset $\displaystyle A \subseteq X$ is determined by deciding whether each of the $\displaystyle n$ elements of $\displaystyle X$ is in the subset: there are two possibilities for each element $\displaystyle x$, namely $\displaystyle x \in A$ and $\displaystyle x \not \in A$. So the total number of possibilities for the subset $\displaystyle A$ is $\displaystyle 2 \times 2 \times \ldots \times 2 = 2^n$. So $\displaystyle |\mathcal{P}(X)| = 2^{|x|}$. Use this to be more rigorous.