Advertisements
Advertisements
प्रश्न
If A and B are subsets of the universal set U, then show that A ⊂ B ⇔ A ∪ B = B
Advertisements
उत्तर
A and B are subsets
To prove: A ⊂ B ⇔ A ∪ B = B
Proof: Let x ∈ A ∪ B
⇒ x ∈ A or x ∈ B
Since, A ⊂ B, we get,
⇒ x ∈ B
⇒ A ⊂ B ⊂ B ......(i)
We know that,
B ⊂ A ∪ B ......(ii)
From equations (i) and (ii),
We get,
A ∪ B = B
Now, Let y ∈ A
⇒ y ∈ A ∪ B
Since, A ∪ B = B, we get,
⇒ y ∈ B
⇒ A ⊂ B
So, A ⊂ B ⇔ A ∪ B = B
Hence Proved.
APPEARS IN
संबंधित प्रश्न
What universal set (s) would you propose for the following:
The set of right 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 \[X = \left\{ 8^n - 7n - 1: n \in N \right\} \text{ and } Y = \left\{ 49\left( n - 1 \right): n \in N \right\}\] \[X \subseteq Y .\]
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, prove that A ⊂ B ⇒ A ∩ 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, prove that:
\[A \cap \left( A \cup B \right) = A\]
Find sets A, B and C such that \[A \cap B, A \cap C \text{ and } B \cap C\]are non-empty sets and\[A \cap B \cap C = \phi\]
Using properties of sets, show that for any two sets A and B,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]
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)
For any two sets A and B, prove that :
\[A' - B' = B - A\]
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 : \[n \left( A \right) = 20, n \left( A \cup B \right) = 42 \text{ and } n \left( A \cap B \right) = 4\] \[n \left( A - B \right)\]
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find: how many can speak Hindi only
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
how many can speak English only.
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,\[A - B =\]
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
B ∪ C ∪ 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 ∩ B) ⊂ A
Let A, B and C be sets. Then show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ 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 none of A, B and C
