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Fearless Symmetry by Robert Gross, Avner Ash

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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.
3
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
4
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.
3
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.
4
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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