Today was only the third lecture of my intro to higher math class, yet the first homework is due in two days and we're required to prove a number of things without even being first instructed on what qualifies as a "proof", or given any idea how to approach them. Unfortunately at the moment there is literally no one I can go to with these questions, so I'm hoping you can help.

Question 1: We've been told that: . Can someone explain why this is the case? Is it just convention?

Question 2: Let P be the statement " for all real numbers x such that ". This statement appears to be true, as the real solution for " " also solves " " But is it required that the "real number x" be a solution to both problems? If I were to change this to: Let P be the statement " for all real numbers x such that ", does the overall statement become false? It seems like a silly question, but how can something that is broken down into a logic dependence also have a second dependence on individual solution sets? For example: Let R be the statement " for all real numbers x such that " Since there are no real solutions for , the statement gets reduced to or . But without an x to plug into , how can it exist in any state, true or false? Isn't it just unknown?

Question 3: Let A and B be non empty sets. Prove that:

This is my pitiful attempt at some kind of "proof":

Let and

Unfortunately that seems a lot more like a "demonstration" than a proof. I have no idea how to approach this.

Question 4: Let A and B be sets. Prove that I understand that a Cartesian product requires the creation of an ordered pair, and the pair is not an ordered pair. As far as structuring a proof to say that, I'm lost.

I apologize for the sloppiness of this post, and for the barrage of questions. If anyone can offer any help or insight, I would very VERY much appreciate it. Thanks in advance.