Question

Let A, B C S. Prove A\ (An B) = (AU B)\ B.

Answer 1

The following proof makes use of logic equivalences.

Explanation:

We have to show that
`x\in A/(A\cap B)` is equivalent to say that
`x\in (A\cup B)/B`. In fact, if
`x\in A/(A\cap B)`, by definition this is equivalent to say that
`x\in A\, \text{and}\, x\\otin A\cap B`, this is equivalent to say that
`x\in A\, \text{and}\, (x\\otin A\, \text{or}\, x\\otin B)`, this is equivalent to
`x\in A\, \text{and}\, x\\otin B`, and this is equivalent to say that
`(x\in A\, \text{or} \, x\in B)\, \text{and}\, x\\otin B`, and this is equivalent to say that
`x\in (A\cup B)/B`.

The Following image can be useful.