Question:
Consider the following statements
i. A = (A ∪ B) ∪ (A − B)
ii. A ∪ (B − A) = (A ∪ B)
iii. B = (A ∪ B) − (A − B)
Which of the statements given above are correct?
निम्नलिखित कथनों पर विचार कीजिए:
i. \(A = (A \cup B) \cup (A – B)\)
ii. \(A \cup (B – A) = (A \cup B)\)
iii. \(B = (A \cup B) – (A – B)\)
उपर्युक्त कथनों में से कौन-से सही हैं?
Options:
- i and iii
- i and ii
- i, ii and iii
- ii and iii
- i और iii
- i और ii
- i, ii और iii
- ii और iii
Correct Answer: d)
Explanation:
i. \((A \cup B) \cup (A – B) = A \cup B \neq A\). Hence statement (i) is incorrect.
ii. \(A \cup (B – A) = A \cup (B \cap A’) = (A \cup B) \cap (A \cup A’) = (A \cup B) \cap U = A \cup B\). Hence statement (ii) is correct.
iii. \((A \cup B) – (A – B) = (A \cup B) \cap (A \cap B’)’ = (A \cup B) \cap (A’ \cup B) = B \cup (A \cap A’) = B\). Hence statement (iii) is correct.
Therefore, statements (ii) and (iii) are correct.
i. \((A \cup B) \cup (A – B) = A \cup B \neq A\)। अतः कथन (i) असत्य है।
ii. \(A \cup (B – A) = A \cup (B \cap A’) = (A \cup B) \cap (A \cup A’) = (A \cup B) \cap U = A \cup B\)। अतः कथन (ii) सत्य है।
iii. \((A \cup B) – (A – B) = (A \cup B) \cap (A \cap B’)’ = (A \cup B) \cap (A’ \cup B) = B\)। अतः कथन (iii) सत्य है।
अतः कथन (ii) और (iii) सही हैं।