$$$$ $$ What mathematics describes is a long standing issue in philosophy. It is hard to break into this subject, but I hope that my intent can be shared. Since I am a mathemtics student who is interested in formal systems, I can’t help to feel nervous when my professor says $A$ is a subset of $B$ if for all $x$, $x\in B$ whenever $x\in A$,

You should know what sets and types are, what a function is, and the consequences of composing functions. In a programming language, you have types that are given to you and types that you can construct with your own definitions. Some examples of these are Int, Bool, and function types Int -> Int -> Int. You may also may have types that generalize over types, an example is the type Optional<T> where T refers to any type.