Advertisements
Advertisements
प्रश्न
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 |
Advertisements
उत्तर
| Column A | Answer |
| (i) ((A′ ∪ B′) – A)′ | (b) A |
| (ii) [B′ ∪ (B′ – A)]′ | (c) B |
| (iii) (A – B) – (B – C) | (a) A – B |
| (iv) (A – B) ∩ (C – B) | (f) (A ∩ C) – B |
| (v) A × (B ∩ C) | (d) (A × B) ∩ (A × C) |
| (vi) A × (B ∪ C) | (e) (A × B) ∪ (A × C) |
Explanation:
(i) ((A′ ∪ B′) – A)′
[(A′ ∪ B′) – A]′
= [(A′ ∪ B′) ∩ A’]’ ......{∵ A – B = A ∩ B’}
= [(A ∩ B)’ ∩ A’]’ ......{∵ (A ∩ B)’ = A’ ∪ B’}
= [(A ∩ B)’]’ ∪ (A’)’
= (A ∩ B) ∪ A
= A
((A′ ∪ B′) – A)′ = A
(ii) [B′ ∪ (B′ – A)]′
[B′ ∪ (B′ – A)]′
= (B’)’ ∩ (B’ – A)’ ......{∵ (A ∪ B)’ = A’ ∩ B’}
= B ∩ (B’ ∩ A’)’ ......{∵ A – B = A ∩ B’}
= B ∩ [(B’)’ ∪ (A’)’] ......{∵ (A ∩ B)’ = A’ ∪ B’}
= B ∩ (B ∪ A)
= B
[B′ ∪ (B′ – A)]′ = B
(iii) (A – B) – (B – C)
Step 1:
A – B
Step 2:
B – C
Step 3:
(A – B) – (B – C)
Clearly, the Venn diagram in Step 3 shows same region as in Step 1
Hence, (A – B) – (B – C) = A – B
(iv) (A – B) ∩ (C – B)
Step 1:
A – B
Step 2:
C – B
Step 3:
(A – B) ∩ (B – C)
Clearly, from the Venn diagram in Step 3:
(A – B) ∩ (B – C) = A ∩ C – B
(v) A × (B ∩ C)
Let y ∈ A × (B ∩ C)
⇒ y ∈ A and y ∈ (B ∩ C)
⇒ y ∈ A and (y ∈ B and y ∈ C)
⇒ (y ∈ A and y ∈ B) and (y ∈ A and y ∈ C)
⇒ y ∈ (A × B) and y ∈ (A × C)
⇒ y ∈ (A × B) ∩ (A × C)
A × (B ∩ C) = (A × B) ∩ (A × C)
(vi) A × (B ∪ C)
Let y ∈ A × (B ∪ C)
⇒ y ∈ A and y ∈ (B ∪ C)
⇒ y ∈ A and (y ∈ B or y ∈ C)
⇒ (y ∈ A and y ∈ B) or (y ∈ A and y ∈ C)
⇒ y ∈ (A × B) or y ∈ (A × C)
⇒ y ∈ (A × B) ∪ (A × C)
A × (B ∪ C) = (A × B) ∪ (A × C)
APPEARS IN
संबंधित प्रश्न
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?
{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?
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 \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]
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:
\[A \cap \left( A \cup B \right) = A\]
Using properties of sets, show that for any two sets A and B,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]
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 the following:
\[A \cap \left( A' \cup B \right) = A \cap B\]
For any two sets A and B, prove the following:
\[A \cap \left( A \cup B \right)' = \phi\]
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 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 in the same universal set. Then,\[A - B =\]
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 ∪ 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 ∪ 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 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 ∪ B = 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.
The set (A ∩ B′)′ ∪ (B ∩ C) is equal to ______.
For all sets A and B, A – (A ∩ B) is equal to ______.
