Answer to Question #85984 in Algebra for RAKESH DEY

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

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