Question

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 1

Answer with Step-by-step explanation:

Let A, B and C are arbitrary sets within a universal set U.

We have to prove that
`( A/B)* C=(A* C)/(B* C)` is always true.

Let
`(x,y)\in (A/B)* C`

Then
`x\in(A/B)` and
`y\in C`

Therefore,
`x\in A` and
`x\\otin B`

Then, (x,y) belongs to
`A* C`

and (x,y) does not belongs to
`B* C`

Hence,
`(x,y)\in(A* C)/(B* C)`

Conversely ,Let (x ,y)belongs to
`(A* C)/(B* C)`

Then
`(x,y)\in (A* C)` and
`(x,y)\\otin (B* C)`

Therefore,
`x\in A,y\in C` and
`x\\otin B,y\in C`

`x\in(A/B)` and
`y\in C`

Hence,
`(x,y)\in(A/B)* C`

Therefore,
`(A/B)* C=(A* C)/(B* C)` is always true.

Hence, proved.