Question

Consider $A = \\{1, 2\\}$ and $B = \\{2, 3\\}$ then which of the following options is correct?

• A. $P\left(A \cup B\right) \ne P\left(A\right) \cup P\left(B\right)$
• B. $P\left(A \cup B\right) = P\left(A\right) \cup P\left(B\right)$
• C. $P\left(A \cup B\right) \ne P\left(A\right) \cap P\left(B\right)$
• D. None of these

Answer 1

Answer: (A)

$P\left(A \cup B\right) \ne P\left(A\right) \cup P\left(B\right)$

Answer 2

Here P\left(A\right) = \left\\{\phi, \left\\{1\right\\}, \left\\{2\right\\}, \left\\{1, 2\right\\}\right\\}, P\left(B\right) = \left\\{\phi, \left\\{2\right\\}, \left\\{3\right\\}, \left\\{2, 3\right\\}\right\\} A \cup B = \left\\{1, 2, 3\right\\} \therefore P\left(A \cup B\right) = \left\\{\phi, \left\\{1\right\\}, \left\\{2\right\\}, \left\\{3\right\\}, \left\\{1, 2\right\\}, \left\\{2, 3\right\\}, \left\\{1, 3\right\\}, \left\\{1, 2, 3\right\\}\right\\} P\left(A\right) \cup P\left(B\right) = \left\\{\phi, \left\\{1\right\\}, \left\\{2\right\\}, \left\\{3\right\\}, \left\\{1, 2\right\\}, \left\\{2, 3\right\\}\right\\} $\therefore P\left(A \cup B\right) \ne P\left(A\right) \cup P\left(B\right)$