1. ## Proof

Show (A U B) X C = (A X C) U (A X B)

And Show that AXC and AXB are disjoint

2. Originally Posted by jzellt
why, yes

Show (A U B) X C = (A X C) U (A X B)
here's a hint. show that $(A \cup B) \times C \subseteq (A \times C) \cup (A \times B)$ and then show $(A \times C) \cup (A \times B) \subseteq (A \cup B) \times C$. this will prove the equality

And Show that AXC and AXB are disjoint
there is no way we can know this without knowing how A, B, and C relate to each other. clearly if B = C or either is a subset of the other, then this is not the case.

and please start using more descriptive titles than just "proof"

3. Originally Posted by jzellt
Show (A U B) X C = (A X C) U (A X B)

And Show that AXC and AXB are disjoint

The above identity is wrong as the following counter example shows:

Let A={1,2},B={3,4} ,C={a}. AND:

AUB= {1,2}U{3,4} = {1,2,3,4}

(AUB)xC = {1,2,3,4}x{a} = {(1,a),(2,a),(3,a),(4,a)}

But:

AxC= {1,2}x{a} = {(1,a),(2,a)}

Also AxB = {1,2}x{3,4} = {(1,3),(1,4),(2,3),(2,4)},and

(AxC)U(AxB)= {(1,a),(2,a)}U {(1,3),(1,4),(2,3),(2,4)}={(1,a),(2,a),(1,3),(1,4) ,(2,3),(2,4)}

Hence: $(AUB)xC\neq(AxC)U(AxB)$

4. Originally Posted by jzellt
Show (A U B) X C = (A X C) U (A X B)

And Show that AXC and AXB are disjoint