write shorthand (Kathy, Pam). It does matter whether you write (Kathy, Pam)
or (Pam, Kathy), doesn't it?
Now we have the first challenge to our claim that all mathematics can be ex-
pressed in set-theoretical terms. Is it possible to have a set that would convey the
same information as the ordered pair (Kathy, Pam)? Of course, {Kathy, Pam} is
no good, because this is the same as {Pam, Kathy}. But there is a trick.
EXERCISE 3(G): Let a, b be any objects. Show that
{{a},{a,b}} = {{c},{c,d}} iff a = c and b = d.
We define (a, 6) = {{a}, {a, b}}, and call (a, b) an ordered pair as opposed to
the unordered pair {a, 6}. What we have done may look artificial. We could have
defined (a, b) the other way round, or perhaps in an entirely different way. And
besides, although Exercise 3 shows that our notion of ordered pair has the desired
properties, can we really say that {{a}, {a, b}} is the ordered pair (a, 6)? Well, in
what sense is (Kathy, Pam) the statement that "Kathy owes Pam some money?"
Only to the extent that you chose it as a symbol for this statement, or rather, for
the relationship concerning debt between Kathy and Pam. The symbol is appro-
priate, since it captures all the relevant information. In the same way, by defining
(a, b) as {{a}, {a, 6}}, we make an agreement on how we shall communicate certain
phenomena in the language of set theory. We are not making any ontological claims,
i.e., we do not stipulate that (a, b) really is {{a}, {a, b}} in some deep philosophical
Given three objects a,b,c, we define the ordered triple (a,b,c) by ((a,b),c).
More generally, for n 2, we define the ordered n-tuple of n objects a
,... , an_i
by (... ((oo, oi),...), a
_i). The Cartesian product of n sets A0,... , An-\ is the
AQ x x An-i = {{do, •.. , a
-i) : °i Ai for each i n}.
EXERCISE 4(PG): (a) Let A,B be nonempty sets. Show that ( J I M x B ) =
(b) Show that in (a) the assumption that A and B are nonempty can be slightly
weakened, but not entirely omitted.
EXERCISE 5(PG): Show that two ordered triples (ao, a±, a2) and (b0, &i, 62) are
equal iff a* = bi for all i 3.
Let us go back to the monetary relationships of your friends Kathy, Pam, John
and Paul. Your list of who owes money to whom may look like this:
(Kathy, Pam), (Kathy, John), (Paul, Pam), (Paul, John).
If F = {Kathy, Pam, John, Paul} is the set of your friends, then their "indebted-
ness relation" is a subset of F x F. More generally, if A is a set, then any subset
R C A x A will be called a binary relation on A. It does not matter whether or
not R represents anything as interesting as indebtedness.
The indebtedness relation may not carry enough information for you. You may
also want to know how much your friends owe each other. An appropriate list may
Previous Page Next Page