10 I The Expanding Universe of Numbers
Let f:IS(m)→Inbe a map such thatf(IS(m))=Inwhich is not injective. Then
there existp,q∈IS(m)withp=qandf(p)= f(q). We may choose the notation
so thatq∈Im.Iffpis a bijective map ofImontoIS(m){p}, then the composite map
h= f◦fpmapsImontoIn. If it is not injective thenm>n,sincem∈M,and
hence alsoS(m)>n.Ifhis injective, then it is bijective and has a bijective inverse
h−^1 :In→Im.Sinceh−^1 (In)is a proper subset ofIS(m), it follows from Proposition 4
thatn<S(m). HenceS(m)∈M.
Propositions 4 and 5 immediately implyCorollary 6For any n∈N,amap f:In→Inis injective if and only if it is surjec-
tive.
Corollary 7If a map f:Im→Inis bijective, then m=n.
Proof By Proposition 4,m<S(n),i.e.m≤n. Replacingfbyf−^1 , we obtain in the
same wayn≤m. Hencem=n.
AsetEis said to befiniteif there exists a bijective map f:E→Infor some
n∈N.Thennis uniquely determined, by Corollary 7. We call it thecardinalityofE
and denote it by #(E).
It is readily shown that ifEis a finite set andFa proper subset ofE,thenFis
also finite and #(F)<#(E).Again,ifEandFare disjoint finite sets, then their union
E∪Fis also finite and #(E∪F)=#(E)+#(F). Furthermore, for any finite setsE
andF, the product setE×Fis also finite and #(E×F)=#(E)·#(F).
Corollary 6 implies that, for any finite setE,amapf:E→Eis injective if and
only if it is surjective. This is a precise statement of the so-calledpigeonhole principle.
AsetEis said to becountably infiniteif there exists a bijective mapf:E→N.
Any countably infinite set may be bijectively mapped onto a proper subsetF,since
Nis bijectively mapped onto a proper subset by the successor mapS. Thus a map
f:E→Eof an infinite setEmay be injective, but not surjective. It may also be
surjective, but not injective; an example is the mapf:N→Ndefined by f( 1 )= 1
and, forn=1,f(n)=mifS(m)=n.
2 IntegersandRationalNumbers
The concept of number will now be extended. The natural numbers 1, 2 , 3 ,...suffice
for counting purposes, but for bank balance purposes we require the larger set...,− 2 ,
− 1 , 0 , 1 , 2 ,...of integers. (From this point of view,−2 is not so ‘unnatural’.) An
important reason for extending the concept of number is the greater freedom it gives
us. In the realm of natural numbers the equationa+x=bhas a solution if and only
ifb>a; in the extended realm of integers it will always have a solution.
Rather than introduce a new set of axioms for the integers, we will define them in
terms of natural numbers. Intuitively, an integer is the differencem−nof two natural
numbersm,n, with addition and multiplication defined by
(m−n)+(p−q)=(m+p)−(n+q),
(m−n)·(p−q)=(mp+nq)−(mq+np).