- #1

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- Homework Statement
- Formulate and prove DeMorgan's Laws for arbitrary unions and intersections.

- Relevant Equations
- 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.

$$A-\bigcup_{B\in\mathfrak{B}} B = \bigcap_{B\in\mathfrak{B}}(A-B)\quad (3)$$

$$A-\bigcap_{B\in\mathfrak{B}} B = \bigcup_{B\in\mathfrak{B}}(A-B)\quad (4)$$

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.