Here is a problem:
Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$.
I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } S_n$. Does it make sense we speak about the relation between $S$ and $\operatorname{Sym}(\Omega)$ when $\Omega$ is an infinite set. Moreover, I know that $|\operatorname{Sym}(\Omega)|=|\Omega|^{|\Omega|}$. How we can reach from $|\Omega|^{|\Omega|}$ to $2^{|\Omega|}$. Thanks.