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: $\displaystyle F \Rightarrow T = T$. Can someone explain why this is the case? Is it just convention?

Question 2: Let P be the statement "$\displaystyle x^2+2=11$ for all real numbers x such that $\displaystyle x^3+32=5$". This statement appears to be true, as the real solution for "$\displaystyle x^3+32=5$" also solves "$\displaystyle x^2+2=11$" 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 "$\displaystyle x^2+2=11$ for all real numbers x such that $\displaystyle x^3+32=6$", 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 "$\displaystyle x^3+32=5$ for all real numbers x such that $\displaystyle x^2+32=0$" Since there are no real solutions for $\displaystyle x^2+32=0$, the statement gets reduced to $\displaystyle F \Rightarrow T = T$ or $\displaystyle F \Rightarrow F = T$. But without an x to plug into $\displaystyle x^3+32=5$, 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: $\displaystyle A \times (B \cup C) = (A \times B) \cup (A \times C)$

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

Let $\displaystyle A = \{a\}, B = \{b\},$ and $\displaystyle C = \{c\}.$

$\displaystyle B \cup C = \{b,c\}$

$\displaystyle A \times (B \cup C) = \{(a,b),(a,c)\}$

$\displaystyle A \times B = \{(a,b)\}$

$\displaystyle A \times C = \{(a,c)\}$

$\displaystyle (A \times B) \cup (A \times C) = \{(a,b),(a,c)\} = A \times (B \cup C)$

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 $\displaystyle A \times \emptyset = B \times \emptyset = \emptyset $ I understand that a Cartesian product requires the creation of an ordered pair, and the pair $\displaystyle \{(a,\emptyset)\}$ 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.