Discrete Math

Question

Prove or disprove the following: for finite sets A and B, $\overline{(A - B) \cup (B - A)} = A \cap B$ . If the proposition is incorrect, do minimal modifications to the same and prove.

Comments

$\overline{(A\cap \bar{B})} \cap \overline{(B \cap \bar{A})}$ = A∩B

$(\bar{A}\cup B) \cap (\bar{B} \cup A)$ = A∩B

Everything without A and having B part ∩ Everything without B and having A part = A∩B

Common to Both is A∩B and Everything except A and B.$\neq$ A∩B

Modification : A $\cup$ B is Universe. then equation hold.

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can we use ven diagram to prove it ?

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yes. you can.

Answer 1

If A Ս B = Universe then the proposition holds

Answer 2

LHS = (A∩B) U (Universe - AUB)

RHS = A∩B

so ...statement is incorrect ..

For statement to be correct, a small modification which can be done is AUB = Universe, that is, A and B are exhaustive ..