1. Objective / Мета роботи #

Implement checks for the fundamental properties of a binary relation on a set, recognise equivalence relations, and generate the inverse of a relation.

By the end of the work the student should be able to:

• represent a binary relation on a set $A$ as a set of ordered pairs $R \subseteq A \times A$;
• test the three core properties — reflexivity ($\forall a,\ (a,a)\in R$), symmetry ($(a,b)\in R \Rightarrow (b,a)\in R$), and transitivity ($(a,b),(b,c)\in R \Rightarrow (a,c)\in R$);
• combine them to decide whether $R$ is an equivalence relation, and explain how such a relation partitions $A$ into equivalence classes;
• construct the inverse relation $R^{-1} = \{(b,a) : (a,b)\in R\}$;
• test antisymmetry and decide whether $R$ is a partial order (reflexive, antisymmetric, and transitive);
• reason about the time complexity of each check.

The work makes the abstract properties of relations concrete and prepares the ground for order relations, functions, and partitions studied later.

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