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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.
asked in Set Theory & Algebra by Veteran (57.4k points) | 115 views
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$\overline{(A\cap \bar{B})} \cap \overline{(B \cap \bar{A})}$ = 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.

2 Answers

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If A Ս B = Universe then the proposition holds
answered by Active (1.4k points)
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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 ..
answered by Active (1k points)


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