Am I correct?

Question

Mathematics

We need to prove the following assertion. Here is the actual proof. Is it right to understand?
Or maybe you saw such a statement in a textbook and can you name it?

Answer 1

If the sets X( ) are arbitrary, then the assertion is false. More precisely, only the first part is true - the right is included in the left.
A = B = {0,1}
X(a,b) = {a xor b}
⋂{a}⋃{b} X(a,b) = [X(0,0) ⋃ X(0,1) ] ⋂ [X(1,0) ⋃ X(1,1)] = {0,1}⋂{0,1} = {0,1}
⋃{b}⋂{a} X(a,b) = [X(0,0) ⋂ X(1,0)] ⋃ [X(0,1) ⋂ X(1,1)] = ∅ ⋃ ∅ = ∅
Your mistake: in (4.1) ∀a ∃b̂(a ), i.e. this b with a roof depends on a. Accordingly, in (5.,1) â depends on b, and there is no way to fix the contradiction.