Recording-2025-02-05T23:10:19.416Z

Translating Sentences into Predicate Logic

Basic Translation

• 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).

More Complex Translation

• 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.

  • Translation:

    • 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).

Explanation of Implication

• 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.