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
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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