Common phrases in proof writing

Question

I am a little confused over the differences between phrases in proof writing. They are: 'Fix an abitrary $x \in S$', 'Fix $x \in S$', 'Let $x \in S$'. I would like to know what each of them means. In addition to this, I'd like further clarification on the two different interpretations of let $x \in S$ (in regards to particular and non-specific $x$'s, whether $x$ is an arbitrary or not constant). If I wanted $x$ to be a specific constant would I still write Let $x \in S$, or is that only for an arbitrary element $x$? When we write $x \in S$, what extra information do we have to give for $x$ to be non-arbitrary and specific?

Answer 1

“Fixing” a value just means to not treat it as changing. An “arbitrary fixed value” may initially seem counterintuitive, but it just means that the value is one specific yet arbitrary value. Essentially, the value has no properties other than its membership in the set, and we are looking at only it. And “let $a$ equal…” simply means to treat the symbol $a$ as a stand-in for what proceeds. Does that clear it up?