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I would appreciate it if I could get some feedback on my attempt to proof De Morgan's Law. That is, if it is correct or if there is a better way to do this proof. I am self-studying probability theory from Grimmett and Stirzaker's book Probability and Random Processes and am going through the exercises. Thanks in advance.

Let ${A_i : i \in I}$ be a collection of sets.

$(\cup_{i} A_{i})^c = (A_1 \cup A_2 \cup A_3 \cup A_4 \ldots)^c$ $= (A_1 \cup A_2)^c \cup (A_3 \cup A_4)^c \cup \dots$ $= (A_{1}^c \cap A_{2}^c) \cup (A_{3}^c \cap A_{4}^c)\ldots$ $=(A_{1}^c \cap A_{2}^c \cap A_{3}^c \cap A_{4}^c \ldots)$ $=\cap_{i} A_{i}^c$

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  • $\begingroup$ How'd you prove that $$\left(A_1\cup A_2\cup B\right)^c=(A_1\cup A_2)^c\cup B^c\;\;??$$ $\endgroup$ – DonAntonio Feb 2 '14 at 13:31
  • $\begingroup$ It's not correct. Your second equality is invalid. Prove by showing each side is a subset of the other. (In one direction: "Let $x$ be an element of the LHS. Do stuff using the definitions... So $x$ is an element of the RHS") $\endgroup$ – David Mitra Feb 2 '14 at 13:31
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Perhaps simpler by definition:

$$x\in\left(\bigcup_i A_i\right)^c\iff x\notin\bigcup_i Ai\iff\;\forall\,i\;,\;x\notin A_i\iff$$

$$\iff \;\forall\,i\,,\;x\in A_i^c\iff x\in\bigcap_i A_i^c$$

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  • $\begingroup$ I follow the reasoning here up to the last "iff". Please can you expand on that. $\endgroup$ – M.K. Feb 2 '14 at 14:02
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    $\begingroup$ @IftikharKhan, an element belongs to the intersection of some sets iff it belongs to each one of them (=to all of them)... $\endgroup$ – DonAntonio Feb 2 '14 at 14:04
  • $\begingroup$ Got it! This is a really nice proof. $\endgroup$ – M.K. Feb 2 '14 at 14:06
  • $\begingroup$ I am really bad with these types of questions (I come from an applied maths background). What is the best way to improve one's ability in this area in your opinion? $\endgroup$ – M.K. Feb 2 '14 at 14:08
  • $\begingroup$ Get the heck out of applied mathematics at once! Just kidding: try to follow 1-2 books on the subject (set theory, discrete mathematics) that appeal to you and do exercises there... $\endgroup$ – DonAntonio Feb 2 '14 at 14:10

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