Suppose π is the power set of the set π = {1,2,3,4,5,6}. For any π β π,β¦
2015
Suppose π is the power set of the set π = {1,2,3,4,5,6}. For any π β π, let |π| denote the number of elements in π and πβ² denote the complement of π. For any π, π β π, let π β π be the set of all elements in π which are not in π . Which one of the following is true?
- A.
βπ β π (|π| = |πβ² |)
- B.
βπ β π βπ β π (|π| = 5, |π| = 5 and π β© π = β )
- C.
βπ β π βπ β π (|π| = 2, |π| = 3 and π β π = β )
- D.
βπ β π βπ β π (π β π = πβ² β πβ² )
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Correct answer: D
Key identity: set difference can be written using complements and intersection.
X \ Y = X β© Y' (because A \ B = A β© B').
Y' \ X' = Y' β© (X')' = Y' β© X.
Since intersection is commutative, X β© Y' = Y' β© X, so X \ Y = Y' \ X' for all X and Y.
Conclusion: The statement asserting X \ Y = Y' \ X' for all X,Y is true; the other given statements are false (see option feedback for short counterexamples or reasoning).
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