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Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws
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[QUOTE="benorin, post: 6348809, member: 37674"] [B]Homework Statement:[/B] Formulate and prove DeMorgan's Laws for arbitrary unions and intersections. [B]Relevant Equations:[/B] DeMorgan's Laws: ##A-(B\cup C) = (A-B)\cap (A-C)\quad (1)## ##A-(B\cap C) = (A-B)\cup (A-C)\quad (2)## So formulating them was easy, just set ##C:=D\cup E## in (1) and set ##C:=D\cap E## in (2) to see the pattern, if ##\mathfrak{B}## is a non-empty collection of sets, the generalized laws are $$A-\bigcup_{B\in\mathfrak{B}} B = \bigcap_{B\in\mathfrak{B}}(A-B)\quad (3)$$ $$A-\bigcap_{B\in\mathfrak{B}} B = \bigcup_{B\in\mathfrak{B}}(A-B)\quad (4)$$ and normally I would proceed to prove these by induction but it said "arbitrary unions and intersections" not countable unions and intersections. So not really sure how to proceed here. Give me a clue please? Also, I used TeX \mathfrak{B} to be my collection of sets, what's the code for the script letters? It doesn't say on the LaTeX help page on PF. [/QUOTE]
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Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws
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