You are in-charge of designing the school curriculum for undergraduate studies and one of your tasks is to arrange the modules in the different trimesters of undergraduate studies. You are given a list of dependencies between modules. For example, MATHS300 depends on MATHS202, MATHS200, MATHS100, MATHS300 and MATHS107 and therefore, it can only be put in a trimester that is after the trimesters for the dependent modules. [For simplicity, you do not have to worry about the number of modules in each trimester (i.e., your undergraduate students can do an arbitrary number of modules in each trimester).] (a)A dependency cycle between modules is a situation where the module dependencies are cyclic. For example, MATH321 depends on MATH213, MATH213 depends on MATH310 and MATH310 depends on MATH321. Prove that there exists a valid schedule of modules to trimesters if and only if there are no dependency cycles. [Update: There are two parts to this proof: Prove that if there is a cycle, you can't schedule the modules in different trimesters. And prove that if there is no cycle, you can always schedule the modules (the number of trimesters can be arbitrary large). You can use any of the lemmas