Let X be a non-empty set and let A, B, C be subsets of X, consider the following statements: 1) A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B) 2) (A ∩ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C 3) (A ∪ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C
1 and 3 only
Concept:
- A ∆ B is the symmetric difference that represents the objects that belong to A or B but not to their intersection.
- A Δ B = (A ∪ B) - (A ∩ B)
- A ⊆ B subset A is a subset of B. set A is included in set B.
- A ⋃ B, ⋃ is aunion, that represents the objects that belong to set A or set B
- A ⋂ B, ⋂ is an intersection, that represents the objects that belong to set A and set B
- a ∈ A means x is an element of A, i.e. belongs to set membership
Calculation:
Given: A, B, C be subsets of X
Statement 1: A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B)
We know that,
If A ∪ B = A ∪ C and A ∩ B=A ∩ C, then B = C which gives
⇒ A ⊂ C
⇒ A ⊂ C ⇒ (A ∩ B) ⊂ (C ∩ B), (A ∪ B) ⊂ (C ∪ B)
Hence statement 1 is true.
Statement 2: (A ∩ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C
Let A = {1, 2, 3}, B = {3, 4, 5} and C = {1, 3, 6, 7, 8}
⇒ A ∩ B = {3} ⊂ (C ∩ B) but A ⊂ C is not true.
Hence, statement 2 is false.
Statement 3: (A ∪ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ C
This is true as (A ∪ B) ⊂ (C ∪ B) for all sets B ⇒ A ⊂ CSimilar Questions
Let S = {2, 4, 6, 8, ______ 20}.
What is the maximum number of subsets does S have?Consider the following statements for the two non-empty sets A and B:
1) (A ∩ B) ∪ (A ∩ B̅) ∪ (A̅ ∩ B) = A ∪ B
2) (A ∪ (A̅ ∩ B̅)) = A ∪ B
Which of the above statements is/are correct?- For any three non-empty sets \(A, B, C\), what is \((A \cup B) - \{(A - B) \cup (B - A) \cup (A \cap B)\}\) equal to ?
If A = {λ, {λ, μ}}, then the power set of A is
If A, B and C are subsets of a given set, then which one of the following relations is not correct?
A set S contains (2n + 1) elements. There are 4096 subsets of S which contain at most n elements. What is n equal to?
If A = { x : x is a multiple of 3} and B = (x : x is a multiple of 4} and C = {x : x is a multiple of 12}, then which one of the following is a null set?
Consider the following statements in respect of sets \(A\) and \(B\):
I. If \(x \in A\) and \(A \in B\), then \(x \in B\).
II. If \(A \subset B\) and \(x \notin B\), then \(x \notin A\).
Which of the statements given above is/are correct?
Consider the following statements:
1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets.
Which of the above statements is/are correct?
Consider three sets X, Y and Z having 6, 5 and 4 elements respectively. All these 15 elements are distinct. Let S = (X - Y) ∪ Z. How many proper subsets does S have?
Important Questions from Set Theory and types of Sets
If A is an open set and B is a closed set, then B - A is
If P and Q be two sets such that P ∪ Q = P, then P ∩ Q will be
A set containing n elements, has exactly ..... subsets.
If P, Q and R are three sets, then which of the following is correct?
Of the members of three athletic teams in a school, 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, 12 play football and cricket and 8 play all the three games. The total number of members in the three athletic teams is:
