Question

prove that (a union b)\(a intersection b)= (a\b) union (b\a) for any two sets a and b in a universal set u.

Answer 1

Let`s assume that:

$$x\in((A\cup B)\setminus(A\cap B))$$

We need to prove that:

$$x\in((A\setminus B)\cup(B\setminus A))$$

$x\in((A\cup B)\setminus(A\cap B)) \iff (x\in(A\cup B))\cap(x\notin(A\cap B)) \iff ((x\in A)\cup(x\in B))\cap((x\notin A)\cup(x\notin B)) \iff ((x\in A)\cap(x\notin A))\cup((x\in A)\cap(x\notin B))\cup((x\in B)\cap(x\notin A))\cup((x\in B)\cap(x\notin B)) \iff \empty\cup((x\in A)\cap(x\notin B))\cup((x\in B)\cap(x\notin A))\cup\empty \iff ((x\in A)\cap(x\notin B))\cup((x\in B)\cap(x\notin A)) \iff (x\in(A\setminus B))\cup (x\in(B\setminus A)) \iff x\in((A\setminus B)\cup(B\setminus A))$

See, that

$$x\in((A\cup B)\setminus(A\cap B)) \iff x\in((A\setminus B)\cup(B\setminus A))$$