Meaning of Well Defined Function

The meaning of a well-defined function

In this article, we are going to understand the meaning of a well-defined function in Mathematics. First we describe what makes a function well-defined and give examples of functions that are well-defined and mappings that are not well-defined. Finally we describe the instances when to check that the function is actually well-defined.

Definition of Well-Defined Function

A mapping is well-defined when given any element of the domain, it is mapped to the same element in the codomain irrespective of its representation.

For example consider a well-defined function $f$ which maps 0.75 into 2 (i.e. $f(0.75)=2$ ). Since the function is well-defined then $f(75%)= 2$ and $f(\frac{3}{4})=2$ and $f(\frac{6}{8})=2$ . This is because since the function is well-defined, and $75%,\frac{3}{4},\frac{6}{8}$ are different representations for $0.75$ , they must all be mapped into the same element $2$ .

On the other hand, if we have a mapping $g$ which maps 0.75 into 4 (i.e. $g(0.75)=4$ ) and 75% into 6 (i.e. $g(75\%)=6$ ), then $g$ is not well-defined because 0.75 and 75% are different representations of the same element but are mapped into distinct elements, namely, 4 and 6.

Is a function always well-defined?

Yes, a function is always well-defined. Hence a mapping must be well-defined in order for it to be a function.

In fact consider the mapping $g$ with $g(0.75)=4$ ) and $g(75\%)=6$ results into a one-to-many mapping which is not a function.

So, strictly speaking, we cannot write or say “a function is not well-defined” because the mapping is not a function in the first place if it is not well-defined! Similarly it does not make complete sense to write “a function is well-defined”, because a function is always well-defined!

An example of a mapping that is not well-defined

Let $h: \mathbb{Q}\rightarrow\mathbb{Z}$ such that $g(\frac{m}{n})=n$ . This mean that the function takes a fraction as its input and output its denominator. Note $\mathbb{Q}$ denotes the set of all real numbers that can be expressed as a fraction, and $\mathbb{Z}$ denotes the set of all integers (i.e. real numbers).

In particular, we have $h(\frac{3}{4})=4$ and $h(\frac{6}{8})=8$ . Since $\frac{3}{4}$ and $\frac{6}{8}$ are different representations of the same number and are mapped to a distinct element, then $h$ is not well-defined.

When to check whether a mapping is well-defined

One must check for well-definition in a mapping, that possibly could also be a function, when there are different representations of some (or all) elements in the domain.

For example in number theory and set theory, $3\mathbb{Z}+2$ is the set obtained by taking any integer multiply it by 3 and adding 2. It results that $3\mathbb{Z}+2=\lbrace \cdots ,-7,-4,-1,2,5,8,\cdots\rbrace$ .

Similarly the set $3\mathbb{Z}+5$ is obtained by taking any integer multiply it by 3 and adding 5. It results
that $3\mathbb{Z}+5=\lbrace \cdots ,-7,-4,-1,2,5,8,\cdots\rbrace$ , which is in fact the same as $3\mathbb{Z}+2$ .

It follows that:

$3\mathbb{Z}+0=\lbrace \cdots,-6,-3,0,3,6,\cdots\rbrace=3\mathbb{Z}+3=3\mathbb{Z}+6=\cdots$

$3\mathbb{Z}+1=\lbrace \cdots,-5,-2,1,4,7,\cdots\rbrace=3\mathbb{Z}+4=3\mathbb{Z}+7=\cdots$

$3\mathbb{Z}+2=\lbrace \cdots,-7,-4,-1,2,5,8,\cdots\rbrace=3\mathbb{Z}+5=3\mathbb{Z}+8=\cdots$

Let $k$ be a mapping defined by $k(3\mathbb{Z}+m)= m$ for all $m\in \mathbb{Z}$ . Here since the elements of the domain (which are sets themselves) have different representations, it makes sense to check about the well-definition property.

In this case, since $k(3\mathbb{Z}+1)= 1$ and $k(3\mathbb{Z}+4)= 1$ , and $3\mathbb{Z}+1=3\mathbb{Z}+4$ , then $k$ is not well-defined.