Discrete Mathematics

258

DISCRETE

MATHEMATICS'--------------------------

Theorem There is

bijection/:

W

~

N.

EXERCISE

Letf(n,

m)

= ((n +

m)

2

+

m-

n) I

2.

(a) List the eleven pairs

of

nonnegative integers (n,

m)

$ 10.

(b) Describe, in general, how given

k E N

2

such that

f(n,

m)

= k.

(c) Verify that f is a bijection.

We will demonstrate that using bijections to specify the size

of

infinite sets yields a sensible theory

of

infinite set sizes - yes, there will

be

more than one size

of

infinity - but it is important to notice from the

start that there will be some dramatic contrasts between size properties

of

finite and infinite sets.

For example, a proper

sul.lset ...