Writing Proof for beginner(s)

I'm glad that some friends and students, outside mathematical field, want to learn the creative way of writing proof. Maybe you can start learning something very basic..like $$ 1+1=2$$. Of cos this sounds so obvious. But how to prove that its true?

Method 1:

$$ Let \, \alpha = 1, then\, (\alpha + \alpha) =$$
$$\alpha(1 + 1) = 2\alpha \Rightarrow$$ By distributive law

You can use any kind of mathematical tools to write proof. Not necessary to be algebra or calculus or geometry etc... writing proof or problem solving is a question that cannot be answered immediately. You will get lost at the very beginning mostly. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.

So let's try using Set as a tool to solve $$1+1=2$$
Method 2:

$$A = \{a_1, a_2, ...\} B = \{b_1, b_2, ...\} C = \{c_1, c_2, ...\} D = \{r,s\}$$
Then, $$ \alpha + \alpha = \Theta(\{A,B\}) = \Theta(\{a_1, a_2, ... b_1, b_2, ...\})$$
$$2\alpha = \Theta(C * D) = \Theta(\{(c_1,r),(c_2,r)....,(c_1,s),(c_2,s)\})$$

But the function $$ f: \{A;B\} \rightarrow \{C * D\}$$
$$f(x) = \biggl \{ \begin{matrix}(c_i, r) \Rightarrow if \, (x = a_i)\\
(c_i, s) \Rightarrow if \, (x = b_i) \end{matrix}$$

is similarly mapping of $$\{A;B\} \, onto \, (C * D)$$
Hence $$\alpha + \alpha = \Theta (\{A;B\}) = \Theta(\{ C * D\}) = 2\alpha$$.
This completes the proof.