Show (A U B) X C = (A X C) U (A X B)
And Show that AXC and AXB are disjoint
Any advice?
why, yes
here's a hint. show that $\displaystyle (A \cup B) \times C \subseteq (A \times C) \cup (A \times B)$ and then show $\displaystyle (A \times C) \cup (A \times B) \subseteq (A \cup B) \times C$. this will prove the equalityShow (A U B) X C = (A X C) U (A X B)
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 Show that AXC and AXB are disjoint
and please start using more descriptive titles than just "proof"
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: $\displaystyle (AUB)xC\neq(AxC)U(AxB)$