Domain of a Variable
The domain of a variable is the set of all possible values a logical or mathematical variable can take such that the expression or equation it appears in is well-defined.
More concretely, it represents the range of values within which we can search for solutions to an equation or determine whether a logical-mathematical statement is true.
What is an open statement?
An open statement is a proposition containing one or more variables that becomes either true or false only when specific values are assigned to those variables.
For instance, consider the following open statement:
$$ x^2 - 4 = 0 $$
This statement is true only for certain values of \( x \). To identify those values, we must first know which values \( x \) is allowed to take - that’s the domain.
In this case, the domain is $ \mathbb{R} $, the set of all real numbers.
Domain vs. Truth Set. These two concepts are often confused, but they serve different purposes. The domain is the complete set of values a variable is allowed to take. The truth set, on the other hand, consists of the specific values that make the statement true. So, the truth set is a subset of the domain. In the equation $ x^2 - 4 = 0 $, the truth set is \( \{ 2, -2 \} \), the actual solutions to the equation.
The same principle applies to logical variables.
Take the following open statement:
“If \( p \), then \( q \)”
Here, \( p \) stands for “it’s raining” and \( q \) for “I take my umbrella.”
The logical variable \( p \) can assume only two values:
- \( V \) (true) → it’s raining
- \( F \) (false) → it’s not raining
The same applies to the variable \( q \):
- \( V \) (true) → I take the umbrella
- \( F \) (false) → I don’t take the umbrella
So, the domain of \( p \) is $ \{V, F\} $, the set of truth values: true and false.
The same domain applies to \( q \): $ \{V, F\} $.
Note. The truth set of the statement “If \( p \), then \( q \)” includes all value pairs that make the implication true. Since there are two variables involved, we can represent the scenario with a truth table: \[
\begin{array}{|c|c|c|}
\hline
\mathbf{p\ (\text{it’s raining})} & \mathbf{q\ (\text{I take the umbrella})} & \mathbf{\text{If } p \Rightarrow q} \\
\hline
V & V & V \\
V & F & F \\
F & V & V \\
F & F & V \\
\hline
\end{array}
\] This table shows that the only scenario in which the implication is false is when \( p = V \) (it’s raining) and \( q = F \) (I don’t take the umbrella). In all other cases, the statement holds. Therefore, the truth set of the statement is $$ \{ (V,V), (F,V), (F,F) \} $$. In other words, if it’s raining, the truth of the statement depends on whether I take the umbrella. But if it’s not raining, the outcome of taking or not taking the umbrella doesn’t affect the validity of the implication. This illustrates a fundamental aspect of conditional statements in propositional logic, where the domain is restricted to the two truth values.
When is it necessary to specify the domain?
In many cases, the domain is understood from context and doesn't need to be stated explicitly. If there are no constraints, it’s generally assumed that the variable ranges over all real numbers.
For example, in the statement $ x^2 - 4 = 0 $, the domain of \( x \) is \( \mathbb{R} \), since there are no limiting conditions.
However, in other contexts, determining the domain carefully is essential - particularly when the statement involves:
- Fractions: because division by zero is undefined.
Consider this open statement: $$ \frac{1}{x - 3} $$ The domain here is all real numbers except 3: $$ \mathbb{R} \setminus \{3\} $$ because \( x = 3 \) would result in a zero denominator, which is not allowed.
- Even roots: because square roots of negative numbers are not defined in the real number system.
Take this example: $$ \sqrt{x - 2} $$ The domain consists of all real numbers greater than or equal to 2: $$ x \geq 2 $$ since the square root is defined only for non-negative inputs.
- Logarithms: because the logarithm of zero or a negative number is undefined.
In this case: $$ \log(x - 1) $$ The domain is all real numbers strictly greater than 1: $$ x > 1 $$ because logarithms are defined only for positive arguments.
Why is defining the domain important?
Identifying the domain is the foundation for analyzing any function or solving any equation.
Overlooking it can lead to mistakes - like accepting values that aren't actually valid solutions.
For example, consider the equation:
$$ \frac{x+2}{x-1} = 0 $$
It’s easy to find that \( x = -2 \) is a solution. But we also need to be aware that \( x = 1 \) is not in the domain, as it would make the denominator zero, and must therefore be excluded from the start.
In short, before working with an equation or exploring a function, always ask yourself: “What values is the variable allowed to take?”
Answering that question ensures your reasoning is mathematically sound, logically coherent, and free from basic but critical errors.
And so on.
