CHAPTER 1

Pairs, Relations, and Functions

EXERCISE 1(G): Consider the sets a = {{0},{{0}}}, 6 = {{{0}},{0}}, and

c = {{{0}}, {0}, {{0}}}- Is any of these sets empty? Are these sets equal, or are

they different? Is 0 an element of a? Or perhaps a subset? How about {0}?

The empty set 0 does not have any element; but this does not prevent it from

being an element of another set, for instance of the set {0}. This set in turn can

be an element of another set; in particular, it is an element of the sets a, b, c of

Exercise 1. Thus, none of these sets is empty.

Are they different? Let us first consider another example. Suppose your in-

structor asked you to determine the union of the set of basketball players and the

set of joggers in your class. Let us assume that John and Mary are joggers, and

Mary is also the only basketball player in your class. You may answer the question

by writing {John, Mary}. Your classmate Bill, who is very fond of Mary, will of

course write {Mary, John}. A more methodical student may write down the set of

joggers followed by the set of basketball players {Mary, John}, {Mary}, and then

erase the middle pair of braces to get the answer {Mary, John, Mary}.

Is any of these answers more correct than the other two? Of course not. They

are just different ways to denote the same set. The first two may be a little more

elegant than the third, but all three are correct. It should be clear from this example

that the order in which the elements of a set are given does not matter, nor does

it matter if elements are repeated. The first property of sets that will be elevated

to the status of an axiom (the so-called Axiom of Extensionality) is the following:

Two sets are equal iff they have the same elements.

Thus, in Exercise 1, a = b = c.

EXERCISE 2(G): Why do we speak of the empty set rather than of an empty

set?

The elements of a are {0} and {{0}}. Since 0 is not equal to any one of these

two, it is not an element of a. However, because 0 has no elements whatsoever,

all elements of 0 are by default also elements of a. As you know, this property is

usually abbreviated by 0 C a. In this book, we shall use the symbol C only if we

want to emphasize that one set is a proper subset of another. We could thus write

0 C a, but it would be wrong to write a C a. On the other hand, {0} is not a

subset of a, since its only element 0 is not a member of the latter.

We have seen that reversing the order of elements of a set does not cause identity

crises for sets. But sometimes order does matter. Suppose your friends often help

each other out in financial difficulties. To keep track of the borrowing that has

been going on, you make a list. Instead of "Kathy owes Pam some money," you

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http://dx.doi.org/10.1090/gsm/008/02