Answer to Question #91342 in Algebra for Ra

Question

Prove that (A∪B)\(A∩B)=(A\B)∪(B\A) for any two sets A and B in a universal set U.

Answer 1

Taking Left hand side (LHS) =

(AUB) \ (A∩B)

Let $x \in$ (A∪B) \ (A∩B)

So $x \in$ (A∪B) $\land$ $x \notin$ (A∩B)

Now for the above condition we have following three options

1. $x \in A \land x \in B$

It means $x \in$ (A∩B). So this contradicts to $x \notin$ (A∩B).

So this is not an option for this point.

2. $x \in A \land x \notin B$

This means $x \in$ A∩B^c . Then $x \in$ (A∩B^c) U (B∩A^c)

So $x \in$ (A\B) ∪ (B\A)

3. $x \notin A \land x \in B$

This means $x\in$  A^c∩B. Then $x \in$ (A∩B^c) U (B∩A^c)

So $x \in$ (A\B) ∪ (B\A)

So LHS = RHS