Let (Ω, F, P) be a probability space. We consider the following statements for A, B ∈ F: (a) If A and B are independent events, then A^c and B^c also independently. (b) For all A and B we have P(A ∩ B) ≤ P(A)P(B). Examine the truth of each statement, i.e., prove it or give a counterexample.   if able provide some explanation with the taken steps, thank you in advance.

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Let (Ω, F, P) be a probability space. We consider the following statements for A, B ∈ F:
(a) If A and B are independent events, then A^c
and B^c also independently.
(b) For all A and B we have P(A ∩ B) ≤ P(A)P(B).
Examine the truth of each statement, i.e., prove it or give a counterexample.

 

if able provide some explanation with the taken steps, thank you in advance.

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