10 CHAPTER 1

There are many kinds of functions, but the most useful ones for

us are the morphisms from a source to a well-understood standard

target. We will call this a representation. It is implicit that the

target we choose is one that we know a lot about, so that from our

knowledge that there is a morphism, and better yet our knowledge

of some additional properties of the morphism, we can obtain new

knowledge about the source object.

DEFINITION: A representation is a morphism from a source

object to a standard target object.

Counting and Inequalities as Representations

Going back to the counting example, we think about ﬁnite sets—

for example, {sun , earth, moon, Jupiter} or {1, Kremlin, π} or any

set that contains a ﬁnite number of items. This collection of ﬁnite

sets contains the special sets {1}, {1, 2}, {1, 2, 3}, and so on. In the

context of counting, given any two ﬁnite sets A and B,amorphism

is a one-to-one correspondence from A to B. A representation in

this case is a morphism from the source (a given ﬁnite set, e.g., the

set of sheep in your ﬂock) to the target, which must be one of the

special sets {1}, {1, 2}, {1, 2, 3}, and so on. The special property that

we demand of the morphisms in the context of counting is that they

should be one-to-one correspondences. For example, if you have a

ﬂock of exactly three sheep for your source, a representation of that

ﬂock must have {1, 2, 3} as its target. Thus, the “essential nature”

of the source that is preserved by the morphism, in this context,is

the number of elements it contains.

There are a lot of possible morphisms—n! to be exact, where

n is the number of elements in the source and target.

5

When we

are counting the number of elements in a set, we do not actually

care about which morphism we grasp onto. But there is no choice

about the target: it is {1, 2, 3, ..., n} if and only if n is the number of

elements in our source.

5

The notation n!, pronounced “n factorial,” means the product of all of the numbers from

1throughn. For example, 5! is 1 · 2 · 3 · 4 · 5 = 120. For an exercise, you can ﬁnd the six

possible one-to-one correspondences from the set {red, blue, green} to the set {1, 2, 3}.

Start Free Trial

No credit card required