Automated teaching system “Sets” (research for organizing the 1st part of the project)

The issues of building an automated learning system “Sets” which will allow students to master one of the important topics of the discipline “Discrete Mathematics” and to develop logical and mathematical thinking in this direction are studied. The corresponding topic of the 1st part of the project includes materials related to the concept of a set, operations on sets, algebra of sets, proofs of statements for sets, and the derivation of formulas for the number of set elements. The system is based on a construction of the statements proof editor for a set and of the formulas derivation editor for the number of set elements, both editors are to be used for teaching. The first of these allows students to split the original statement into a number of simpler statements, taken together equivalent to the original statement, to choose a method of proving each simple statement and to conduct their step-by-step proof. The second editor allows (using the inclusion-exclusion principle and the formula of the number of complement elements) to derive a step-by-step formula for the number of set elements through the specified numbers of elements for sets from which the resulting set is constructed.
An important part of the system is to monitor the correctness of all actions of students, and on this basis the entire learning system is developed. The logical supervision over the correctness of the selected action in the first editor is performed by a Boolean function created by the system and corresponding to this action and by checking it for identical truth. In the second editor, invariants such as characteristic strings of the set and of its number of elements are used for verification. The rest of the system is related to learning of set algebra and to preparation to editors usage. The main focus here is on the learning strategy in which testing the understanding of the learned material is rather rigorous and eliminating the random choice of answers. The division of the material into sections with verification of the success of teaching not only by tests, but also by exercises and tasks, allows students to master the complex logical and mathematical techniques of proving statements for sets and derivation of formulas for the number of set elements.