I am a computer science major currently self-studying mathematics.
Today I was studying topology and found out that I don't even know elementary facts like $(A \times B) \subset (C \times D)$ doesn't imply $A \subset B$ and $C \subset D%$.
I want to make my proof rigorous, but it is not easy. Do math majors also make mistakes like above? And if they don't, Can I be like them without undergraduate education in mathematics? How?
Thanks in advance.