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?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

LohaHridoy

4 views

2025-01-04 04:42

1 Answers

AhmedNazir

Option 1 : 1 only

Concept:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Calculation:

Let, us suppose the universal set U = A ∪ B.

Statement 1:

⇒ (A ∩ B) ∪ (A ∩ B̅) ∪ (A̅ ∩ B) = {A ∪ (A ∩ B̅)} ∩ {B ∪ (A ∩ B̅)} ∪ (A̅ ∩ B)

⇒ (A ∩ B) ∪ (A ∩ B̅) ∪ (A̅ ∩ B) = {(A ∪ A) ∩ (A ∪ B̅)} ∩ {(A ∪ B) ∩ (B ∪ B̅)} ∪ (A̅ ∩ B)

⇒ (A ∩ B) ∪ (A ∩ B̅) ∪ (A̅ ∩ B) = {A ∩ U} ∩ {U ∩ U} ∪ (A̅ ∩ B)

⇒ (A ∩ B) ∪ (A ∩ B̅) ∪ (A̅ ∩ B) = A ∪ (A̅ ∩ B) = (A ∪ A̅) ∩ (A ∪ B) = A ∪ B

Hence statement 1 is Correct.

Statement 2:

⇒ (A ∪ (A̅ ∩ B̅)) = (A ∪ A̅) ∩ (A ∪ B̅)

⇒ (A ∪ (A̅ ∩ B̅)) = U ∩ (B ∩ A̅) = A̅ ∩ B

Hence, statement 2 is wrong.

4 views Answered 2022-09-04 04:29

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