Algebra Know-It-ALL

outside of the rectangle and part of it is the outside of the oval. Any element inside the region

bounded by the dashed line is counted only once.

Are you confused?

Once more, go back and look at Fig. 2-1, again noting that the set of all the women in Chicago is a proper

subset of the set of all the people in Illinois, that is, Cw⊂Ip. The diagram also makes it plain that the union

ofCw with Ip is just Ip. To be in one set or the other (or both), a person only has to be a resident of Illinois,

that is, an element of Ip. It’s not necessary to be a woman, and it’s not necessary to be in Chicago. Here’s

how you would write that:

Cw∪Ip=Ip

Here’s a challenge!

Can you find two sets of whole numbers, with one of them infinite, but such that their union contains

only a finite number of elements?

Solution

Don’t think about this for too long. You’ll never find two such sets! An element in the union of two sets

only has to belong to one of the sets. If a set has infinitely many elements, then the union of that set with

any other set—even the null set—must have infinitely many elements as well.

Practice Exercises

This is an open-book quiz. You may (and should) refer to the text as you solve these problems.

Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not

represent the only way a problem can be figured out. If you think you can solve a particular

problem in a quicker or better way than you see there, by all means try it!

1. Is there any set that is a subset of every other set? If so, what is it? If such a set can’t
   exist, why not?


2. Continuing with the theme of Problem 1, is there a way to take nothing and build up
   an unlimited number of different sets from it? If so, show an example. If not, explain
   why not.


3. What set does the small, dark-shaded triangle marked P represent in Fig. 2-6? What set
   does the dark-shaded, irregular, four-sided figure marked Q represent?


4. If you consider all the possible intersections of two sets in Fig. 2-6, which of those
   intersection sets are empty?


5. Is the universal set a subset of itself? Is it a proper subset of itself?


6. Give an example of two sets, both with infinitely many elements, but such that one is a
   proper subset of the other.

Practice Exercises 33