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Finite and Infinite Set

Finite and Infinite Set

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Finite and Infinite Set | Mathematics Tutorials - 2 #pep #math #learnmat...fini

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Infinite I

We already defined what `finite' means and hence also what `infinite' signifies. A set X is infinite if there is no natural number n such that X and {i:0≤i<n} have the same size.

The notion of `same size' needs a better definition than in Finite I. We say that two sets, X and Y, have the same size is we can couple the elements of X and Y in a `monogamous' way: every x in X is coupled with exactly one y from Y and, conversely, every y in Y is coupled to exactly one x from X.

For example, set of months in a year and the set of provinces in The Netherlands have the same size; here is a coupling: {(January,Groningen), (February,Drente), (March,Friesland), (April,Overijssel), (May,Flevoland), (June,Gelderland), (July,Utrecht), (August,Noord-Holland), (September,Zuid-Holland, (October,Zeeland), (November,Noord-Brabant), (December,Limburg)}. We usually write such a coupling as a set of ordered pairs.

The set of pairs (n,n+1), where n runs through the set N of natural numbers shows that N={0,1,2,3,…} and {1,2,3,…} (N except 0) have the same size. Similarly, using the pairs (n,2n), we can see that the set of even natural numbers has the same size as the full set N.

Many familiar infinite sets have the property that we illustrated above: they have the same size as some proper subset. The interval [0,1] has the same size as the interval [0,1) for example: make the pairs (2-n,2-n-1) for n in N and (x,x) for all other x in [0,1]. It is a good exercise to verify that this match-making effort does the job.

Dedekind used this property `the same size as a proper subset' as his definition of infinite sets and the examples above appear to confirm that this is a good definition. The problem is, however, that it is not quite equivalent to our definition. A next time we shall see why that is.

In mathematical jargon a the kind of coupling that we discussed here is called a bijection. It pays to study this notion carefully; it is indispensable if you want to talk sensibly about finite and infinite sets.