Let A, B, and C be arbitrary sets within a universal set, U. For each of the following statements, either prove that the statement is always true or show a counterexample to prove it is not always true. When giving a counterexample, you should define the three sets explicitly and say what the left-hand and right-hand sides of the equation are for those sets, to make it clear that they are not equal.(A \ B) × C = (A × C) \ (B × C)

Answer with Step-by-step explanation:Let A, B and C are arbitrary sets within a universal set U.We  have to prove that [tex]( A/B)\times C=(A\times C)/(B\times C)[/tex] is always true.Let [tex](x,y)\in (A/B)\times C[/tex]Then [tex] x\in(A/B) [/tex] and [tex] y\in C[/tex]Therefore, [tex] x\in A[/tex] and [tex] x\notin B[/tex]Then, (x,y) belongs to [tex] A\times C[/tex]and (x,y) does not belongs to [tex] B\times C[/tex]Hence,[tex] (x,y)\in(A\times C)/(B\times C)[/tex]Conversely ,Let (x ,y)belongs to [tex] (A\times C)/(B\times C)[/tex]Then [tex] (x,y)\in (A\times C)[/tex] and [tex] (x,y)\notin (B\times C)[/tex]Therefore,[tex] x\in A,y\in C[/tex] and [tex] x\notin B,y\in C[/tex][tex] x\in(A/B)[/tex] and [tex]y\in C[/tex]Hence, [tex] (x,y)\in(A/B)\times C[/tex]Therefore,[tex] (A/B)\times C=(A\times C)/(B\times C)[/tex] is always true.Hence, proved.