Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given: Total number of students = 200
Number of students study Mathematics = 120
Number of students study Physics = 90
Number of students study Chemistry = 70
Number of students study Mathematics and Physics = 40
Number of students study Mathematics and Chemistry = 50
Number of students study Physics and Chemistry = 30
Number of students study none of them = 20
Let U be the total number of students, P, M and C be the number of students study Physics, Mathematics and Chemistry respectively
To find: Number of students who study all the three subjects n(M ∩ P ∩ C)
n(U) = 200
n(M) = 120
n(P) = 90
n(C) = 70
n(M ∩ P) = 40
n(M ∩ C) = 50
n(P ∩ C) = 30
Number of students who play either of them = n(P ∪ M ∪ C)
= Total – None of them
= 200 – 20
= 180 ........(i)
Number of students who play either of them = n(P ∪ M ∪ C)
= n(C) + n(P) + n(M) – n(M ∩ P) – n(M ∩ C) – n(P ∩ C) + n(P ∩ M ∩ C)
= 120 + 90 + 70 – 40 – 30 – 50 + n(P ∩ M ∩ C)
= 160 + n(P ∩ M ∩ C) .......(ii)
From (i) and (ii)
160 + n(P ∩ M ∩ C) = 180
⇒ n(P ∩ M ∩ C) = 180 – 160
⇒ n(P ∩ M ∩ C) = 20
Hence, there are 20 students who study all three subjects.
APPEARS IN
संबंधित प्रश्न
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}
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}
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
B ⊂ A ∪ B
For any two sets A and B, prove that
A ∩ B ⊂ A
For any two sets A and B, show that the following statements are equivalent:
(i) \[A \subset B\]
(ii) \[A \subset B\]=ϕ
(iii) \[A \cup B = B\]
(iv) \[A \cap B = A .\]
For three sets A, B and C, show that \[A \cap B = A \cap C\]
For three sets A, B and C, show that \[A \subset B \Rightarrow C - B \subset C - A\]
For any two sets of A and B, prove that:
\[A' \cup B = U \Rightarrow A \subset B\]
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.
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 :
\[A' - B' = B - A\]
For any two sets A and B, prove the following:
\[A - \left( A - B \right) = A \cap B\]
For any two sets A and B, prove the following:
\[A - B = A \Delta\left( A \cap B \right)\]
Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( A - B \right)\]
Let A and B be two sets such that : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( B - A \right)\]
In a survey it was found that 21 persons liked product P1, 26 liked product P2 and 29 liked product P3. If 14 persons liked products P1 and P2; 12 persons liked product P3 and P1 ; 14 persons liked products P2 and P3 and 8 liked all the three products. Find how many liked product P3 only.
Let A and B be two sets in the same universal set. Then,\[A - B =\]
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 A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find
A ∪ B ∪ D
If X and Y are subsets of the universal set U, then show that Y ⊂ X ∪ Y
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 ∪ B = B
If A and B are subsets of the universal set U, then show that (A ∩ B) ⊂ A
A, B and C are subsets of Universal Set U. If A = {2, 4, 6, 8, 12, 20} B = {3, 6, 9, 12, 15}, C = {5, 10, 15, 20} and U is the set of all whole numbers, draw a Venn diagram showing the relation of U, A, B and C.
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.
If A = {1, 3, 5, 7, 9, 11, 13, 15, 17} B = {2, 4, ..., 18} and N the set of natural numbers is the universal set, then A′ ∪ (A ∪ B) ∩ B′) is ______.
For all sets A and B, A – (A ∩ B) is equal to ______.
Match the following sets for all sets A, B, and C.
| Column A | Column B |
| (i) ((A′ ∪ B′) – A)′ | (a) A – B |
| (ii) [B′ ∪ (B′ – A)]′ | (b) A |
| (iii) (A – B) – (B – C) | (c) B |
| (iv) (A – B) ∩ (C – B) | (d) (A × B) ∩ (A × C) |
| (v) A × (B ∩ C) | (e) (A × B) ∪ (A × C) |
| (vi) A × (B ∪ C) | (f) (A ∩ C) – B |
