Example Sentence: Every student in this class has taken a course in Java.
Step 1: Define the Domain
Choose the set of all students in this class as the domain.
Step 2: Define the Predicate
Let j(x) denote "x has taken a course in Java."
Translation:
This can be represented as ∀x j(x) (For all x, x has taken a course in Java).
Modified Domain: Assume the domain is all people.
Step 1: Define Additional Predicate
Let s(x) denote "x is a student in this class."
Step 2: Clarify Variables
Although the domain is all people, we need to specify the context of students in this class.
Step 3: Combine Predicates with Universal Quantifier
The translation now requires combining predicates:
s(x): signifies x is a student in this class.
j(x): signifies x has taken a course in Java.
This can be represented as ∀x (s(x) → j(x)) (For all x, if x is a student in this class, then x has taken a course in Java).
Understanding the Implication:
The translation uses the logical implication (→) to indicate a conditional relationship.
Rephrased: For all x, if x is a student in this class, then x has taken a course in Java.
This ensures we are using both predicates correctly within the context of a universal quantifier.