Munkres Topology ch1 ex #9 - Generalized DeMorgan's Laws

[benorin]

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.

 

[andrewkirk]

Do it by looking at individual elements.
First prove that any element of the set described by the LHS of the equation must be an element of the RHS.
Then do the reverse. That will show equality of the two sides.
For the first part, take an element of the LHS and identify all logical statements that are true of it, re membership of the various sets. Then use those statements to show that the element must be in the set described by the RHS.
You will need to use logical quantifiers $\forall$ and maybe also $\exists$.

 

[Infrared]

As @andrewkirk suggested, show that the two sets on either side of the supposed equalities (3,4) contain each other, and conclude that they must be equal.

normally I would proceed to prove these by induction but it said "arbitrary unions and intersections" not countable unions and intersections.

You would only be able to use induction for $\textbf{finite}$ collections of sets anyway, not countable.

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.

Try \mathcal or \mathscr instead of \mathfrak.