Domain of a function is all the elements of the first set?

Question

I am reading about functions in the textbook "Discrete Mathematical Structures" by Kolman et.al. They have given in an example that \begin{equation} A=\{1,2,3\} \quad\text{and}\quad B=\{x,y,z\} \end{equation}

Then $R=\{(1,x),(2,x)\}$ is a function.

But a google search revealed that this is not a function since 3 is not involved.

Whether the book is wrong or I got something wrong

Answer 1

At least in the third edition of Kolman et al., function is defined as follows:

Let $A$ and $B$ be nonempty sets. A function $f$ from $A$ to $B$, which is denoted $f:A\to B$, is a relation from $A$ to $B$ such that for all $a\in\operatorname{Dom}(f)$, $f(a)$ contains just one element of $B$. Naturally, if $a$ is not in $\operatorname{Dom}(f)$, then $f(a)=\varnothing$. If $f(a)=\{b\}$, it is traditional to identify the set $\{b\}$ with the element $b$ and write $f(a)=b$.

This obviously depends on some prior definitions and conventions relating to relations. They define a relation $R$ from $A$ to $B$ in the usual way, as a subset of $A\times B$, and immediately introduce the notation $a\,R\,b$. The domain of $R$ is then defined (in words) to be $$\operatorname{Dom}(R)=\left\{a\in A:\exists b\in B\big(a\,R\,b\big)\right\}\;,$$ and examples make it clear that $\operatorname{Dom}(R)$ need not be all of $A$. Finally, for $a\in A$ they define $R(a)=\{b\in B:a\,R\,b\}$.

If $A=\{1,2,3\}$, $B=\{x,y,z\}$, and $R=\{\langle 1,x\rangle,\langle 2,x\rangle\}$, then certainly $R\subseteq A\times B$, so $R$ is a relation from $A$ to $B$. $\operatorname{Dom}(R)=\{1,2\}$, $R(1)=\{x\}$, and $R(2)=\{x\}$, so it’s true that $f(a)$ contains just one element of $B$ for each $a\in\operatorname{Dom}(R)$. Thus, $R$ does indeed satisfy their definition of function from $A$ to $B$.

However, what they call a function from $A$ to $B$ is more commonly called a partial function from $A$ to $B$, i.e., a function from a subset of $A$ to $B$. Many people reserve the term function from $A$ to $B$ for total functions from $A$ to $B$, i.e., functions whose domain is all of $A$.

Answer 2

There are so many concepts in mathematics, and not all has a unique name that means exactly the same thing in every context.

Most probably in that specific book it is well explained what is meant by 'function' (within the book!) However, if the domain of a function is 'not full', it is usually called partial function. Considering partial functions as 'primary' notions indeed can have some advantages which are probably used in the book.

Answer 3

It is possible that the book uses some non-standrad convention or definition. The standard approach to functions is that a function must associate one, and only one, value to every element of the domain.