1. ## Well defined

1.) Show addition in the reals is well defined. i.e. $x=y\implies x+a=y+a$ for $x,y,a\in\mathbb{R}$.

2.) Given a set $S$, choose $(x,y), (w,z)\in S\times S$.

Show $(x,y)=(w,z)\iff x=w$ and $y=z$.

2. In both of these questions I'm not sure what we're taking for granted. In the first case, the proof will depend on how you've constructed the real numbers. In the second case, I have always taken this fact as the definition of the "=" relation between elements in the cross-product, so I'm not sure what there is to prove. Perhaps there is something to prove if you've defined pairs as sets of sets.

3. Have you defined $(a,b)$ as $(a,b) = \{a, \{a,b \} \}$?

4. This is Kazimierz Kuratowski's definition.

5. Yes, it is, but the question is, does mathman88 use that definition?