Intuitively, why does A, B independent $\iff$ A, $B^C$ independent $\iff A^C, B^C$ independent?
Proposition 2.5.3.
If A and B are independent, then A and $B^C$ are independent, $A^C$ and B are independent, and $A^C$ and $B^C$ are independent.
Blitzstein, Introduction to Probability (2019 2 ed) p 64.
I'm seeking solely intuition. I'm NOT asking about how to prove these independences, because I already linked to a proof that I grasped. Proposition 2.5.3 isn't intuitive because
☣ 2.5.8. It is easy to make terrible blunders stemming from confusing independence and conditional independence. Two events can be conditionally independent given E, but not independent given $E^C$. Two events can be conditionally independent given E, but not independent. Two events can be independent, but not conditionally independent given E.
Op. cit. p 65.
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If events $A$ and $B$ are independent, then the probability that event $A$ happens is not affected by whether $B$ happens. If it isn't affected by whether $B$ happens, then it isn't affected by whether $B$ doesn't happen. (The rest of your statements can be shown to be true with the same reasoning.)
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