Another important operation on sets is “taking complements.” Definition If E and F are sets, then E  F œ ÖB − E À B  F× is called the complement of F in E . If it is clearly understood that we are taking the complement of F in some particular set E, then not µ bother to write the “E” and simply write F - or F in place of E  FÞ For example, in a discussion µ where Y is the universal set and F © Y , our textbook writes F to mean the same thing as Y  F . A Venn diagram schematically shows E  F À

Examples ‘   œ  Ðthe set of irrational numbersÑ. (There is no standard mathematical name for the set of irrational numbers. Personally, I like using , because it leads to the nice-looking equation    œ ‘Þ)   Ö8 −  À $l8× œ Ö"ß #ß %ß &ß (ß )ß "!ß ""ß ÞÞÞ× œ the set of all natural numbers that are not divisible by $.

There are a few other important properties of unions, intersection and complements. But before we mention them, we want to generalize so that we will be able to take the union or intersection of more than just two sets  even infinitely many.

Definition If T is any collection of sets, then -T œ ÖD À D − E for some set E − T× In words, -T œ the set of “all members of members of T .” If then

T œ ÖE" ß E# , E$ ×, where E" œ Ö"ß #×ß E# œ Ö#ß $ß %×ß E$ œ Ö$ß (×, - T œ Ö"ß #ß $ß %ß (×, which is the same as ÐE"  E# Ñ  E$ œ E"  ÐE#  E$ Ñ

As we said earlier, the use of different letter styles is just for convenience. In the preceding example, we could have used all lower case letters, writing Let A œ ÖBß Cß D× where B œ Ö"ß #×ß C œ Ö#ß $ß %× and D œ Ö$ß (×Þ Then -A œ Ö? À ? − B or ? − C or ? − D× œ œ Ö? À ? − Ö"ß #× or ? − Ö#ß $ß %× or ? − Ö$ß (×× œ Ö"ß #ß $ß %ß (×Þ Similarly, we can define the intersection of any collection of sets.

Definition If T is any collection of sets, then +T œ ÖD À D − E for every E − T×

In words, +T is the set of “all elements that are in every member of T .” For the collection defined above, +T œ +A œ g

Sometimes it's convenient to attach indices Ð œ “labels”Ñ to the sets in a collection of sets. To do this, we use some set ? an indexing set ( œ “set of labels”) Definition Suppose sets W! are given, one for each ! − ?. Then we say that the collection T œ ÖW! À ! − ?× is indexed by ?. In that case, we might also write T more informally as ÖW! ×!−? or even merely as ÖW! × if the indexing set ? is clearly understood by everyone. When we have an indexed collection such as T œ ÖW! À ! − ?×, then we write unions and intersections in a variety of ways. For example, -T œ -ÖW! À ! − ?× œ -!−? W!

+T œ +ÖW! À ! − ?× œ +!−? W!

œ ÖB À Ðb! − ?Ñ B − W! × œ ÖB À Ða! − ?Ñ B − W! ×

When the particular indexing set ? is understood or irrelevant, we might even skip writing “! − ?” and just write -! W! or +! W! . If the indexing set is , then T œ ÖW" ß W# ß ÞÞÞß W8 ß ÞÞÞ× we might also write _ -T œ -8− W8 œ -8œ" W8 _ +T œ +8− W8 œ +8œ" W8

œ ÖB À Ðb8 − Ñ B − W8 × œ ÖB À Ða8 − Ñ B − W8 ×

Examples (Look at each one carefully to be sure you understand the notation. A few of them may take a little thought to check the final result.) 1) For each B − Ò!ß _Ñß let MB œ the interval Ò!ß BÓ of real numbers. Then T œ ÖMB À B − Ò!ß _Ñ× is an infinite collection of closed intervals, one for each B   !Þ Here, the indexing set ? is Ò!ß _Ñ. -T œ -B ! MB œ Ò!ß _Ñ +T œ +B 0 MB œ Ö!×

Ðwhy?Ñ Ðwhy?Ñ

2) Let F8 œ Ò!ß "  8" Ó © ‘Þ Then ´ œ ÖF" ß F# ß ÞÞÞß F8 ß ÞÞÞ× is an infinite collection of closed intervals on the real line, indexed by Þ _ -T œ -8− F8 œ -8œ" F8 œ Ò!ß #Ó _ +T œ +8− F8 œ +8œ" F8 œ Ò!ß "Ó

3) If E8 is the closed interval  

1 n

,1 

1‘ n on

Ðwhy?Ñ Ðwhy?Ñ the real line, then

-ÖE8 À 8 − × œ -_ 8œ" E8 œ Ò  "ß "Ñ Ðwhy only a “half-closed” interval?Ñ +ÖE8 À 8 − × œ +_ 8œ" E8 œ Ö!×

4) If F8 œ Ò8ß _Ñ © ‘ß then - F8 œ Ò"ß _Ñ and + F8 œ g _

_

8œ"

8œ"

5) Let c ÐEÑ be the power set of EÞ Then -c ÐEÑ œ E and +c ÐEÑ œ gÞ

It's possible to generalize properties about set operations (unions, intersections and complements) to operations with infinite families of sets. For example, we said earlier that unions are associative ÐE  FÑ  G œ E  ÐF  GÑ. This can be generalized as follows: Suppose we have three indexed families ÖE! À ! − W× and ÖE! À ! − X × and ÖE! À ! − [ ×. Then Ð-!−WX E! Ñ  Ð-!−[ E! Ñ œ Ð-!−W E! Ñ  Ð-E!−X [ Ñ

Example A relatively simple fact to prove is: ÐG  HÑ  ÐI  J Ñ œ ÐG  IÑ  ÐGY J Ñ  ÐH  IÑ  ÐH  J Ñ

ЇÑ

(Try it!) A generalized version of the same result for indexed families of sets is the following: Ð+ !−E W! Ñ  Ð+ " −F W" Ñ œ + !−Eß "−F ÐW!  W" Ñ

Ї‡Ñ

Be sure you see that if Ї‡Ñ is true, then Ð‡Ñ must automatically be true. Here is a proof of the more general result Ї‡ÑÞ As usual, we argue that the left-hand side (LHS) is a subset of the right-hand side (RHS) in the proposed equation, and then make the argument in the opposite direction. If B − LHS, then B − + !−E W! or B − + " −F W" . If B − + !−E W! , then Ða! − EÑ B − W! Þ Therefore Ða! − EÑÐa" − BÑ B − W!  W" , so B − + !−Eß"−F ÐW!  W" Ñ œ RHS.

If B − + " −F W" , then Ða" − FÑ B − W" Þ Therefore Ða! − EÑÐa" − BÑ B − W!  W" so B − + !−Eß"−F ÐW!  W" Ñ œ RHS. To prove the other “half”, that RHS © LHS, we use an indirect proof (contraposition)

If B  LHS œ (+ !−E W! Ñ  Ð+ " −F W" Ñ, then B  + !−E W! and B  + " −F W" . Then Ðb! − EÑ B  W! , say B  W!! . Similarly, Ðb" − FÑ B  W" , say B  W"! Þ Therefore B  W!!  W"! , so B  + !−Eß "−F ÐW!  W" Ñ œ RHS.

Therefore RHS œ LHS. ñ The example illustrates a choice that sometimes has to be made in writing mathematics: do you prove the “most general” version of something that you possibly can (like (**)) ? or do you prove something simpler and easier to understand (like (*)) ? The answer depends on your purpose. In writing a lower level mathematics text, one usually states a theorem that is no more general that what's going to be needed in that particular course  so that it's easier to understand. In doing research, a mathematician likes to prove as general a result as s/he possibly can. After all, who knows when, in the future, that extra generality might turn out to be helpful? In that setting, the attitude is never “discard” any knowledge. The next theorem connects complements with unions and intersections. Theorem (DeMorgan's Laws) Suppose E is a set and that ÖW" À " − F× is an indexed collection of sets. Then 1) E  -ÖW" À " − F× œ +ÖE  W" À " − F×, 2) E  +ÖW" À " − F× œ -ÖE  W" À " − F×

and

In words, 1) says that “the complement of a union is the intersection of the complements” and 2) says that “the complement of an intersection is the union of the complements.” In the simplest cases: what does the theorem say about E  ÐF  GÑ and about E  ÐF  GÑ? In the situation where we are taking all complements within a given universe Y , what does the theorem say about ÐF  GÑ- and ÐF  GÑ- ? Proof 1) We have to show that LHS œ RHS. We could do the proof in two parts (as in the last theorem): show first that LHS © RHS, and then that RHS © LHS. But sometimes we can shorten the write-up of this kind of “iff” proof by laying out an argument where every new statement is logically equivalent to the preceding one. In other words, each step in the argument is “reversible” (iff). Below, you can read the argument “from top to bottom” to prove LHS © RHS and read it “from bottom to top” to prove RHS © LHS. B − LHS œ E  -ÖW" À " − F× Í B − E and B  -ÖW" À " − F× Í B − E and B  W" for every index " Í B − E  W" for every index " Í B − +ÖE  W" À " − F× œ RHS.

The proof of part 2) of DeMorgan's Laws is similar and is left as an exercise. ñ

Why are 1) and 2) referred to as “DeMorgan's Laws”  a name we already used in discussions of logic? In fact DeMorgan's Laws (for sets) are really just a rephrasing into set theory of DeMorgan's Laws (in logic). Example Suppose T ÐBÑ and UÐBÑ are open sentences, and that some universe Y is given. Let E œ ÖB − Y À T ÐB) is true× and F œ ÖB − Y À UÐBÑ is true×. Then

E  F œ ÖB − Y À T ÐBÑ ” UÐBÑ is true× and E  F œ ÖB − Y À T ÐBÑ • UÐBÑ is true×, and

DeMorgan's Laws (as stated above, for sets) give us that Y  ÐE  FÑ Å Òthe set of B's for which µ ÐT ÐBÑ ” UÐBÑÑ is trueÓ

œ

ÐY  EÑ  ÐY  FÑ Å Òthe set of B's for which µ T ÐBÑ • µ UÐBÑ is trueÓ

that is