Question

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

Answer 1

Answer:

The following proof makes use of logic equivalences.

Step-by-step 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 $x\in A\, \text{and}\, x\notin A\cap B$, this is equivalent to say that$x\in A\, \text{and}\, (x\notin A\, \text{or}\, x\notin B)$, this is equivalent to $x\in A\, \text{and}\, x\notin B$, and this is equivalent to say that $(x\in A\, \text{or} \, x\in B)\, \text{and}\, x\notin B$, and this is equivalent to say that $x\in (A\cup B)/B$.

The Following image can be useful.