Question
Download Solution PDFWhich of the following statements is/are true:
- A ∪ ϕ = A
- A ∪ A = A
- A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
- A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
For any set A, B and C the following property are true:
- Idempotent law: A ∪ A = A
- Identity law: A ∪ ϕ = ϕ ∪ A = A
- Distributive law: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Calculation:
Statement 1 : A ∪ ϕ = A
We know that, statement 1 is true because according to identity law A ∪ ϕ = ϕ ∪ A = A
Statement 2: A ∪ A = A
We know that, statement 2 is true because according to idempotent law A ∪ A = A
Statement 3: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
We know that, statement 3 is true because according to distributive law A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Statement 4: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
We know that, statement 4 is true because according to distributive law A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Hence, all the statements are true.
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