Advertisements
Advertisements
Question
If A and B are subsets of the universal set U, then show that A ⊂ B ⇔ A ∪ B = B
Solution
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
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?
{0, 1, 2, 3, 4, 5, 6}
If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then prove that:
\[\left( A \cup B \right)' = A' \cap B'\]
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 three sets A, B and C, show that \[A \subset B \Rightarrow C - B \subset C - A\]
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,\[\left( A \cup B \right) \cap \left( A \cap B' \right) = A\]
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.
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:
\[A - \left( A - B \right) = A \cap B\]
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 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 U be the universal set containing 700 elements. If A, B are sub-sets of U such that \[n \left( A \right) = 200, n \left( B \right) = 300 \text{ and } \left( A \cap B \right) = 100\].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
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 ∩ Y = X
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 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 ______.
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 ______.
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 |
