For finite sets, cardinality of a set = number of elements in the set.

Two infinite sets have the same cardinality if their elements can be placed in 1-to-1 correspondence with each other.

The cardinality of N is denoted (aleph-null).

Any set of cardinality is called countable.

Is Z countable ? Yes. e.g. :

Surely Q isn't countable ? Yes it is !

e.g. imagine the whole set of rationals in an infinite array :

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Traverse this array with a series of diagonal V's from the middle of the top row downwards. Ignore fractions not in their lowest terms ( and , n > 1). We get :

.

N	1	2	3	4	5	6	7	8	9	10	11	....
Q	0	1	-1	2		-	-2	3		-	-3	....

- and eventually every member of Q appears in the list. So Q is countable.

Maybe every infinite set is countable !

What about R ?

R is uncountable Proof

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