A set is enumerable (or denumerable or countable or listable) if it can be written as
. In other words, there is a one-to-one correspondence between the set and the positive integers.
When there is a one-to-one correspondence between two sets
and
, the sets are said to be equipotent (or equinumerous or equipollent or of the same cardinality), which is denoted by
. The Demonstration shows that
.
Let
be the set of integers. The mapping that matches
with the
term of the sequence
is a one-to-one correspondence, so
.
Use square paper to draw pictures showing that
and
. In other words, for
draw a path that passes through each lattice point above the
axis exactly once. The path does not have to be connected. Similarly, for
, find a path that passes through every lattice point in the plane exactly once.
It is not true that every infinite set is countable. For example, neither the real numbers
nor the complex numbers
are not countable (but
).
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