NCERT Class XI Mathematics - Sets - Solutions
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Question : 64
Total: 73
Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.
Solution:
No, it is not true.
Take A = {1, 2} and B = {2, 3}
Then A ∪ B = {1, 2, 3}
⇒ P(A) = {f, {1}, {2}, {1, 2}} and
P(B) = {f, {2}, {3}, {2, 3}}
∴ P(A) ∪ P(B) = {f, {1}, {2}, {3}, {1, 2}, {2, 3}} .....(i)
A ∪ B = {1, 2, 3}
⇒ P(A ∪ B) = {f, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}} .....(ii)
From (i) and (ii), we have
P(A ∪ B) ≠ P(A) ∪ P(B).
Take A = {1, 2} and B = {2, 3}
Then A ∪ B = {1, 2, 3}
⇒ P(A) = {f, {1}, {2}, {1, 2}} and
P(B) = {f, {2}, {3}, {2, 3}}
∴ P(A) ∪ P(B) = {f, {1}, {2}, {3}, {1, 2}, {2, 3}} .....(i)
A ∪ B = {1, 2, 3}
⇒ P(A ∪ B) = {f, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}} .....(ii)
From (i) and (ii), we have
P(A ∪ B) ≠ P(A) ∪ P(B).
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