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

• benorin
In summary, the generalized laws (3) and (4) state that for a non-empty collection of sets, the difference of a set and the union (or intersection) of the collection is equal to the intersection (or union) of the set and the complement of each element in the collection. These laws can be proven by showing that the two sets on either side of the equation contain each other. Induction cannot be used for these proofs as they involve arbitrary unions and intersections, not just finite ones. The code for script letters in LaTeX is \mathcal or \mathscr, instead of \mathfrak.
benorin
Homework Helper
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.

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##.

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.

benorin said:
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.

benorin said:
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.

## 1. What is Munkres Topology ch1 ex #9 about?

Munkres Topology ch1 ex #9 is about Generalized DeMorgan's Laws, which are mathematical laws that describe the relationship between complement and union/intersection operations in set theory.

## 2. What is the significance of Generalized DeMorgan's Laws?

Generalized DeMorgan's Laws are important in set theory as they provide a way to express the complement of a union/intersection in terms of the individual complements of the sets involved.

## 3. Can you provide an example of Generalized DeMorgan's Laws?

One example of Generalized DeMorgan's Laws is: (A ∪ B)' = A' ∩ B', where A and B are sets. This means that the complement of the union of two sets is equal to the intersection of the individual complements of the sets.

## 4. How are Generalized DeMorgan's Laws useful in topology?

In topology, Generalized DeMorgan's Laws are useful in proving theorems and solving problems involving complements, unions, and intersections of sets. They also help to simplify complex set expressions.

## 5. Are there any exceptions to Generalized DeMorgan's Laws?

Yes, there are exceptions to Generalized DeMorgan's Laws, such as when dealing with infinite sets or when the sets involved are not disjoint. In these cases, the laws may not hold true.

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