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I have the following scenario:

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As you can see, I have two classes that are Lecturers and Students. The class Teacher Assistants are a mix between Lectures and Students, that is because they can enrolled into courses, but they can also lecture some basic topics (without being considered Lecturers). I came with the idea to model this situation using Interfaces because I will program in in Java. Is this modelling correct?

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So that the TA class will implement the Interface Teaches, which contains an array of the courses assigned to this student to teach.

But if I model in that way I realize that I am loosing the class Lecturers at all. How I can model this situation of multiple inheritance, but not loosing the class Lecturers? I mean if I program Lecturers as an interface it would not have any methods that I would need further, for example, calculus of its wage benefits and so on. Any recommendation?

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You can make Lecturer into an interface, then have both TeachingAssistant and Professor implement it. TeachingAssistant can extend Student, because from a logical point of view, teaching assistants are students.

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But Logically student needn't have any properties from lecturer. Better would be to have a different interface for students. TA would implement both students and teachers interface.
That's why Lecturer is just an interface.
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One option is to have Teacher and Teachable interfaces and then have Lecturer implement Teacher (for want of a word like Teacherable!), Student implement Teachable and TA implement both Teacher and Teachable.

Lecturer, Student and TA could all extend Faculty_Staff.

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tbodt's approach, in my opinion, is generally sound. You just have to ask: if some client code needs a Lecturer, would an instance of a TA suffice? If there's an issue with that, you can choose to design it with composition, abstracting the ability to teach into a separate class of its own, something like TeachingJob (I can't think of a better name). This way the only thing that actual Professors and TeachingAssistants share is that they have similar teaching jobs; they do not belong to the same "class" (interface actually).

Hybrid approach would be to involve TeachingJob with the Lecturer interface, e.g. Lecturer would have a assign(TeachingJob assignment) method.

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