Truth Set

The truth set is the subset of the domain consisting of all values that make an open statement true.

When dealing with an open statement - that is, a proposition containing at least one variable (such as "x > 5") - we can't immediately determine whether it’s true or false.

Its truth depends entirely on the value assigned to the variable (or variables, if there’s more than one).

For this reason, it’s essential to distinguish between two key ideas: the domain and the truth set.

The domain is the set of all possible values a variable can take. Think of it as the "universe" within which the variable operates.
For example, take the open statement: $$ x > 5 $$ If we specify that x is a natural number, then the domain is ℕ.
The truth set is the subset of the domain that includes only the values that make the statement true.
For instance, given the statement $$ x > 5 $$ the truth set consists of the natural numbers that satisfy "x > 5". These are: 6, 7, 8, 9, 10, 11, … So, the truth set is: $$ \{ x \in ℕ \mid x > 5 \} $$

In short, the truth set tells us exactly when an open statement turns into a true proposition.

We construct it by starting with the domain and filtering out all values that don’t satisfy the condition.

A Practical Example

Let’s consider the open statement:

$$ x^2 = 9 $$

Suppose the domain of the variable $ x $ is the set of integers ℤ.

To determine the truth set, we look for integer values of x that make the equation “x² = 9” true.

In this case, the solutions are x = 3 and x = -3. Therefore, the truth set is:

$$ \{ x \in ℤ \mid x = 3 \ \text{or} \ x = -3 \} $$

Example 2

The concept of a truth set also applies to conditional statements, such as:

"If it rains, I take an umbrella."

Here, the implicit variable is the weather - that is, whether it rains or not. Let’s represent this variable as $ p $.

The resulting action (“I take an umbrella”) is the outcome, or $ q $.

p = "it rains"
q = "I take an umbrella"

Both variables have a domain consisting of two truth values: true and false.

According to propositional logic, a conditional statement of the form "If p, then q" is false in only one scenario: when p is true (it rains), but q is false (I don’t take the umbrella).

In every other case, the statement is logically considered true.

\[
\begin{array}{|c|c|c|}
\hline
p & q & p \rightarrow q \\
\hline
\text{T} & \text{T} & \text{T} \\
\text{T} & \text{F} & \text{F} \\
\text{F} & \text{T} & \text{T} \\
\text{F} & \text{F} & \text{T} \\
\hline
\end{array}
\]

For example, if it rains ($ p = \text{T} $) and I take the umbrella ($ q = \text{T} $), the statement is true.

If it rains ($ p = \text{T} $) but I don’t take the umbrella ($ q = \text{F} $), the statement is false. So far, so clear.

However, what often goes unnoticed is that the statement remains logically true even when it doesn’t rain ($ p = \text{F} $), regardless of whether I take the umbrella or not. Since the condition never occurs, it can't be violated - so the entire conditional holds true.

Note. In everyday language, we tend to interpret “If it rains, I take an umbrella” as also implying its converse: “If it doesn’t rain, I don’t take an umbrella.” But in formal logic, that’s not the case. The conditional statement concerns only the situations in which the condition is actually met.

Therefore, the truth set of this conditional includes all scenarios in which $ p \rightarrow q $ is true, namely:

\[ \{ (p = \text{T}, q = \text{T}),\ (p = \text{F}, q = \text{T}),\ (p = \text{F}, q = \text{F}) \} \]

In plain terms, the statement is true in every case except when it rains and I don’t take the umbrella.

And so on.