Question

We define the following predicates: C(x) is " x is a computer science student" M(y) is " y is a mathematics course" T(x,y) is "student x has taken course y " Write each of the following propositions as logical expressions using the predicates defined above, quantifiers, and logical connectives, where the domain of variable x is all KU students, and the domain of variable y is all courses offered by KU. 1. Every mathematics course has been taken by some computer science student.

Answer 1

The proposition 'every mathematics course has been taken by some computer science student' is written in logical expressions as ∀y (M(y) → ∃x (C(x) ∧ T(x, y))), using universal and existential quantifiers.

Step-by-step explanation:

To represent the proposition that every mathematics course has been taken by some computer science student using logical expressions, we use predicates, quantifiers, and logical connectives. The statement can be formulated in predicate logic as follows:

∀y (M(y) → ∃x (C(x) ∧ T(x, y)))

This expresses that for all courses y, if y is a mathematics course, then there exists at least one student x, such that x is a computer science student and student x has taken course y. The use of universal and existential quantifiers (∀ for 'for all', and ∃ for 'there exists', respectively) is crucial in formalizing these types of universal affirmative statements or conditionals that stipulate necessity and sufficiency within a logical framework.