Hey there! In doing practice exercises for topology, I find that I can reason through them easily enough to find the correct answer. The problem is that I rely predominantly upon verbal logic in order to make the proofs work. How does one go about proofs using only the math notation?

A simple example: Prove that $\displaystyle A\in 2^A$

Another simple example: Prove that $\displaystyle A \subset B \iff A \cup\B = B$

thanks for your time.