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
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