6 CHAPTER 1

To make this mathematically precise, we make two deﬁnitions:

DEFINITION: A set is a collection of things, which are called

the elements of the set.

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For example, the collection of all odd numbers is a set, and the

odd number 3 is an element of that set.

DEFINITION: A one-to-one correspondence from a set A to a

set B is a rule

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that associates to each element in A exactly

one element in B, in such a way that each element in B gets

used exactly once, and for exactly one element in A.

Digression: Deﬁnitions

A mathematician uses the term “deﬁnition” in a way that might be

surprising to nonmathematicians. The Oxford English Dictionary

deﬁnes “deﬁnition” as “a precise statement of the essential nature

of a thing.” Mathematicians agree that a deﬁnition should be

“precise,” but we are not so sure about capturing the “essential

nature.” Our deﬁnition of one-to-one correspondence above will let

you recognize a one-to-one correspondence if one is shown to you.

Suppose that A is the set {red, blue, green} and B is the set {1, 2, 3}.

Then a o ne-to-one correspondence between the two sets is given by

red → 1

blue → 2

green → 3.

You can check that this associates to each element of the set A a

different element of the set B, and that each element of the set B is

used once.

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A set may be described by listing all of its elements between curly braces, so that {1, a, b}

is the set with the three elements 1, a,andb. A set may also be described using a qualiﬁer

preceded by a colon, so that {x : x > 0andx is real} is the set of all positive real numbers.

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By “rule,” we mean any deﬁnite means of association. It need not be given by a formula.

For example, it can be given by a list that tells which sheep in the morning and in the

evening were counted by the same pebble.

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