For each of the following for all statements either disprove

For each of the following “for all” statements, either disprove it with a specific counterexample, or prove it using one of the techniques we’ve studied.  If you want to prove it, be sure to clearly indicate what you’re assuming, and what your goal is.

1) For all sets A,B, if there exists a set C such that   then A = B.

2) For all sets A,B, if for all sets C,   then A = B.

3) For all sets A,B, if there exists a set C such that A and   , then A = B.

4) For all Boolean functions F,G,H of degree 2,if F(x,y) + H(x,y) = G(x,y) + H(x,y) for all x and y, then F(x,y) = G(x,y) for all x and y.

Solution

For all sets A,B, if there exists a set C such that A and , then A = B.