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I ran across the following set identity in a probability problem: $$(E_1 E_2 E_3)^c = E_1 \cup (E_1 E_2^c) \cup (E_1 E_2 E_3^c)$$ and can't make heads or tails of it. I tried applying De Morgan's law as follows: $$ (E_1 E_2 E_3)^c = E_1^c \cup E_2^c \cup E_3^c$$ but don't see how this helps. I am not necessarily looking for a proof of the identity, but rather a way to understand heuristically why it is true, although a proof would probably help with that.

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A consequcne of your identity is $$(E_1\cap E_2 \cap E_3)^c \subseteq E_1;$$ do you see a problem?

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    $\begingroup$ Yes, I see the contradiction here. Thank you, the solution I was looking at came from a less than completely reputable source and I won't be as trusting the next time. $\endgroup$
    – Is12Prime
    Mar 14, 2019 at 0:52

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