Advertisements
Advertisements
Question
Is it true that for any sets A and \[B, P \left( A \right) \cup P \left( B \right) = P \left( A \cup B \right)\]? Justify your answer.
Advertisements
Solution
\[\text{ False } . \]
\[\text{ Let } X \in P\left( A \right) \cup P\left( B \right)\]
\[ \Rightarrow X \in P\left( A \right) or X \in P\left( B \right)\]
\[ \Rightarrow X \subset A or X \subset B\]
\[ \Rightarrow X \subset \left( A \cup B \right)\]
\[ \Rightarrow X \in P\left( A \cap B \right) \]
\[ \therefore P\left( A \right) \cup P\left( B \right) \subset P\left( A \cup B \right) . . . \left( 1 \right)\]
\[\text{ Again }, \text{ let } X \in P\left( A \cup B \right)\]
\[\text{ But } X \not\in P\left( A \right) \text{ or } x \not\in P\left( B \right) \left[ \text{ For example let } A = \left\{ 2, 5 \right\} \text{ and } B = \left\{ 1, 3, 4 \right\} \text{and take } X = \left\{ 1, 2, 3, 4 \right\} \right]\]
\[So, X \not\in P\left( A \right) \cup P\left( B \right)\]
\[\text{ Thus }, P\left( A \cup B \right) \text{ is not necessarily a subset of }P\left( A \right) \cup P\left( B \right) .\]
APPEARS IN
RELATED QUESTIONS
What universal set (s) would you propose for the following:
The set of isosceles triangles.
Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?
Φ
Given the sets, A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, the following may be considered as universal set (s) for all the three sets A, B and C?
{1, 2, 3, 4, 5, 6, 7, 8}
If \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cup B \right)' = A' \cap B'\]
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cap B \right)' = A'B' .\]
For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A
For three sets A, B and C, show that \[A \cap B = A \cap C\]
For any two sets, prove that:
\[A \cup \left( A \cap B \right) = A\]
For any two sets, prove that:
\[A \cap \left( A \cup B \right) = A\]
If A and B are sets, then prove that \[A - B, A \cap B \text{ and } B - A\] are pair wise disjoint.
Show that for any sets A and B, A = (A ∩ B) ∪ ( A - B)
Each set X, contains 5 elements and each set Y, contains 2 elements and \[\cup^{20}_{r = 1} X_r = S = \cup^n_{r = 1} Y_r\] If each element of S belong to exactly 10 of the Xr's and to eactly 4 of Yr's, then find the value of n.
For any two sets A and B, prove that :
\[A' - B' = B - A\]
Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( A - B \right)\]
A survey shows that 76% of the Indians like oranges, whereas 62% like bananas. What percentage of the Indians like both oranges and bananas?
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
how many can speak English only.
Let A and B be two sets that \[n \left( A \right) = 16, n \left( B \right) = 14, n \left( A \cup B \right) = 25\] Then, \[n \left( A \cap B \right)\]
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
B ∪ D
If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ B ∪ C
If X and Y are subsets of the universal set U, then show that Y ⊂ X ∪ Y
If X and Y are subsets of the universal set U, then show that X ⊂ Y ⇒ X ∩ Y = X
If A and B are subsets of the universal set U, then show that A ⊂ A ∪ B
If A and B are subsets of the universal set U, then show that (A ∩ B) ⊂ A
Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three subjects.
In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers. Find the number of families which buy newspaper A only.
Given the sets A = {1, 3, 5}. B = {2, 4, 6} and C = {0, 2, 4, 6, 8}. Then the universal set of all the three sets A, B and C can be ______.
For all sets A and B, A – (A ∩ B) is equal to ______.
