our-shoulders method, when we are done we will know whether
the sacks contained the same number of potatoes or not, but the
place will be strewn with potatoes and we will not know what that
number is. If instead we use counting words, we can count the
potatoes one sack at a time, neatly, and then compare the answers.
The Deﬁnition of a Representation
A one-to-one correspondence is an example of a function and of a
morphism. We will be using these terms throughout this book. We
will take a stab at deﬁning them now, and reﬁne and amplify the
deﬁnitions as we continue.
DEFINITION: A function from a set A to a set B is a rule that
assigns to each element in A an element of B.Iff is the name
of the function and a is an element of A,thenwewritef (a)to
mean the element of B that is assigned to a. A function f is
often written as f : A → B.
DEFINITION: A morphism is a function from A to B that
captures at least part of the essential nature of the set A in
its image in B.
We must be intentionally vague in this chapter about the way
that a morphism “captures the essential nature” of A,mostly
because it depends on the nature of the entities A and B.Whenwe
use the word “morphism” later in the book, our source A and target
B will both be groups. After we have deﬁned “group” in chapter 2,
we will revisit the idea of a “morphism of groups” in chapter 12.
Some people may think “morphism” is an ugly word, but it is
the standard mathematical term for this concept. The longer word
“homomorphism” is also used, but we will stick with the shorter
version. It derives from the Greek word for “form,” and we view the
“essential nature” captured by a morphism as the “form” of A.