Propositional Functions

Propositional Functions

1. Definition

A propositional function is an expression that contains one or more variables and becomes a proposition when specific values are substituted for those variables.

It is usually written as:

Where:

• $P$ is the propositional function (or predicate)
• $x$ is a variable
• $P(x)$ is not a proposition until a value is assigned to $x$

2. Example

Let $P(x)$: “x is an even number”

• $P(2)$: True
• $P(5)$: False
• $P(x)$: Not a proposition until $x$ is specified

So, $P(x)$ becomes a statement only after assigning a value to $x$

3. Domain of Discourse

The domain of discourse (or universe of discourse) is the set of all possible values that the variable can take.

Example:

If the domain is $\mathbb{Z}$ (integers), and
$P(x)$: “x < 0”

Then:

• $P(-1)$: True
• $P(3)$: False
• $P(x)$: Propositional function until $x$ is assigned

4. Multiple Variables

A propositional function can have more than one variable.

Example:

Let the domain of both $x$ and $y$ be real numbers.

• $Q(4, 2)$: True
• $Q(1, 5)$: False

5. Truth Set

The truth set of a propositional function is the set of all values from the domain that make the statement true.

Example:

Let $P(x): x^2 = 4$, with domain $\mathbb{Z}$

Then the truth set is:

6. Use in Predicate Logic

Propositional functions are the building blocks of predicate logic, where variables are quantified using quantifiers like:

• Universal quantifier: $\forall x \, P(x)$: “For all x, P(x) is true”
• Existential quantifier: $\exists x \, P(x)$: “There exists an x such that P(x) is true”

These convert propositional functions into propositions.

7. Application

Propositional functions are used to:

• Express statements involving variables
• Define mathematical sets
• Build logical arguments in predicate logic
• Analyze truth conditions in mathematics, computer science, and formal reasoning