Our deﬁnition of one-to-one correspondence, however, does not
tell you the “essential nature” of a one-to-one correspondence. We
have given you no clue why you should care about one-to-one
correspondences, nor does our deﬁnition tell you how to make a
Even when a mathematical deﬁnition technically has all of the
properties listed by the OED , it often strikes nonmathematicians as
unusual. A mathematical deﬁnition can redeﬁne a commonly used
word to mean something else. For example, mathematicians refer
to “simple” groups, which are in fact not particularly simple. They
deﬁne the words “tree” and “quiver” in ways that have nothing to
do with oaks and arrows.
Sometimes a mathematician deﬁnes an object in terms of its
properties, and only then proves that an object with these prop-
erties exists. Here is an example: The greatest common divisor of
two positive integers a and b can be deﬁned to be a positive number
d so that:
1. d divides a.
2. d divides b.
3. If c is any other number that divides both a and b,thenc
With this deﬁnition, it is not obvious that the greatest common
divisor exists , because there might not be any number d that
satisﬁes all three properties. So right after making the deﬁnition,
it should be proved that a number with the properties outlined
In our example, each pebble corresponded to one sheep in the
morning and one sheep in the afternoon. This sets up the rule
that associates to each morning sheep the afternoon sheep that
shared its pebble. This rule is a one-to-one correspondence under
the conditions of our story.
But we do not need to know any set theory, nor what a one-to-
one correspondence is, to count sheep in this way. In fact, we do