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)

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lublana lublana · Answer 1 · 2019-08-05T10:18:38+02:00

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.