Question

tutoring maria sometimes stays after school to tutor classmates. if it is tuesday, then marla tutors chemistry. if marla tutors chemistry, then she arrives home at 4 p.m. today maria arrived home at 4 p.m. can it be concluded that today is tuesday? explain your reasoning.

Explanation:

We analyze the logical structure. Let's define: \( T \): It is Tuesday, \( C \): Marla tutors chemistry, \( A \): Marla arrives home at 4 P.M. We have the implications \( T
ightarrow C \) and \( C
ightarrow A \). By the law of syllogism, \( T
ightarrow A \). But we know \( A \) (Marla arrived home at 4 P.M.), and in logic, the converse of \( T
ightarrow A \) (i.e., \( A
ightarrow T \)) is not necessarily true. Just because the consequent of an implication is true, we can't conclude the antecedent is true. So arriving home at 4 P.M. ( \( A \) ) doesn't guarantee it's Tuesday ( \( T \) ), as there could be other days when she tutors chemistry (or other reasons she arrives at 4 P.M.).

Answer:

No, it cannot be concluded that today is Tuesday. The given implications show that if it is Tuesday, she arrives home at 4 P.M. ( \( T
ightarrow A \) ), but the reverse (if she arrives at 4 P.M., then it is Tuesday, \( A
ightarrow T \)) is not a valid inference. There could be other circumstances (e.g., she tutors chemistry on other days, or other activities lead to her arriving at 4 P.M.) that make \( A \) true without \( T \) being true.