The Expression Well-Defined in Mathematics
Terminology
In mathematics, the expression “well-defined” is used when a mathematical object can be identified in an unambiguous, objective, and clear manner.
Explanation
Well-defined functions
The phrase “well-defined” is mainly used for functions. For two sets $X$ and $Y$, a function from $X$ to $Y$ is a relation that assigns to each element of $X$ exactly one element of $Y$. That is, a well-defined function is one that satisfies the following two conditions.
- For every element $x$ of the domain $X$, the function value $f(x)$ exists.
- If $x_{1} \sim x_{2}$, then $f(x_{1}) = f(x_{2})$.
Here $\sim$ is an equivalence relation. In other words, even if the same object is represented differently, the corresponding function values must be equal. Let us look at concrete examples in which each condition is violated.
- For $X = [0, 1]$ and $Y = \mathbb{R}$, if we define $f : X \to Y$ as $f(x) = \dfrac{1}{x}$, then the value $f(0)$ does not exist, so $f$ is not a well-defined function.
- For $X = \mathbb{Q}$ and $Y = \mathbb{R}$, let $f : X \to Y$ be $f(\frac{a}{b}) = a$. Then we have $\frac{1}{2} = \frac{3}{6}$, but the corresponding function values are not equal.
$$ f(\textstyle \frac{1}{2}) = 1 \ne 3 = f(\frac{3}{6}) $$ Therefore such $f$ is not a well-defined function.
Well-defined operations
If you form a ternary expression using a binary operation that does not satisfy the associative law, the operation is not well-defined. For example, since addition and multiplication satisfy associativity, when written as below the result does not change regardless of whether you perform the first or the second operation first.
$$ 1 + 2 + 3 = 6, \qquad 2 \times 3 \times 4 = 24 $$
However, in the case of division, as in the example below, the result depends on which operation is performed first.
$$ \frac{1}{8} \overset{?}{=} 1 \div 2 \div 4 \overset{?}{=} 2 $$
Therefore the above expression is not well-defined; it is better to use parentheses to indicate the intended meaning unambiguously or to use multiplication symbols only.
$$ \frac{1}{8} = (1 \div 2) \div 4, \qquad 1 \div (2 \div 4) = 2 $$
Well-known trolling examples such as $48 \div 2(9 + 3) = ?$ can also be regarded as ill-defined problems, so debating what the answer is is a waste of time. Put the parentheses correctly and move on.