Translate The Following Sentences From English To Predicate

Translate the following sentences from English to predicate logic. The domain that you are working over is X, the set of people. You may use the functions S(x), meaning that “x has been a student of 6.042,” A(x), meaning that “x has gotten an ‘A’ in 6.042,” T(x), meaning that “x is a TA of 6.042,” and E(x, y), meaning that “x and y are the same person. It is to translate several statements about people, students, and TAs into predicate logic.

(a) There are people who have taken 6.042 and have gotten A’s in 6.042.

(b) All people who are 6.042 TAs and have taken 6.042 got A’s in 6.042.

(c) There are no people who are 6.042 TAs who did not get A’s in 6.042.

(d) There are at least three people who are TAs in 6.042 and have not taken 6.042.

Sample Paper For Above instruction

In formal logic, translating English statements about people, students, and TAs into predicate logic allows for clear and unambiguous analysis of the statements' logical structure. Given the domain of discourse as the set of all people, with functions and predicates defined as S(x), A(x), T(x), and E(x, y), translating these sentences involves expressing existence, universality, and negation accurately.

Part (a) states that there exist some individuals who have taken 6.042 and have achieved A grades. This is formally expressed as: ∃x (S(x) ∧ A(x)). This formula claims that there is at least one person x such that x has been a student of 6.042 and x has received an A in the course.

Part (b) asserts that every person who is a TA and has taken 6.042 has received an A. The formalization is: ∀x ((T(x) ∧ S(x)) → A(x)). This indicates that for all individuals x, if x is a TA and has taken 6.042, then x has an A grade in 6.042.

Part (c) indicates that no person who is a TA and has taken 6.042 did not receive an A. This is the negation of the existence of such individuals, formalized as: ¬∃x ((T(x) ∧ S(x)) ∧ ¬A(x)), or equivalently, ∀x ((T(x) ∧ S(x)) → A(x)). Note that this formulation aligns with part (b), asserting a universal condition.

Part (d) specifies that there are at least three individuals who are TAs and have not taken 6.042. This can be formalized as: ∃x∃y∃z (x ≠ y ∧ y ≠ z ∧ x ≠ z ∧ T(x) ∧ T(y) ∧ T(z) ∧ ¬S(x) ∧ ¬S(y) ∧ ¬S(z)). This ensures that among the TAs, at least three distinct individuals have not taken the course, emphasizing the existence of such a group.

Expressing these sentences in predicate logic offers a precise way to analyze and reason about the relationships and properties within the domain of people regarding their course enrollment and grades. These formalizations are foundational in logic-based approaches to knowledge representation and reasoning in computer science and artificial intelligence.

References

  • Hinrichs, T. (2011). Logic for Computer Science. Springer.
  • Leighton, T., & Dijk, M. van. (2010). Mathematics for Computer Science. MIT OpenCourseWare.
  • Enderton, H. B. (2001). A Mathematical Introduction to Logic. Academic Press.
  • Huth, M., & Ryan, M. (2004). Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press.
  • Russell, S., & Norvig, P. (2010). Artificial Intelligence: A Modern Approach. Prentice Hall.

Related Assignments