A predicate is a symbol that represents a property or a relation. It can also be thought of as an statement that can evaluate to either true or false depending on it's inputs. Predicates are usually represented by capital letters (such as , , and ), and the inputs are called variables and are usually lower case letters (such as , , and ).

For example, the predicate could represent the property "x is an animal", where is a variable that can be replaced by specific values, such as "dog" or "cat". In this case, would evaluate to true where as would evaluate to false.

In terms of relations, predicates could be used to represent statements such as "x is greater than y", in this case the predicate takes two variables and could be written as . Again, the variables in these predicates can be substituted with specific values to determine the truth value of the statement.

Domain

The domain of a predicate is the set of all possible inputs or variables for which the predicate can be evaluated. For example, the predicate = "x is a prime number" has a domain of all integers.

Finding Truth Values of a Predicate

To find a truth value of a predicate simply evaluate the statement with the given inputs replacing the variables in the predicate. The set of all inputs that make the predicate true is called the truth set.

Example

Let be the predicate "" with the domain being all real numbers. Evaluate , , and .