91Ó°ÊÓ

Set theory is a fundamental part of mathematics that deals with collections of objects, known as sets. In set theory, a set is a well-defined collection of distinct objects, which can range from numbers to letters, or even other sets.

Each object in a set is called an element, and sets are usually denoted by capital letters, while elements are denoted by lower-case letters. A key point in set theory is the way sets relate to each other, and one of the ways to describe these relationships is through binary relations.

Binary relations are a way to formalize how the elements of two sets relate to one another. They consist of pairs of elements, where one element comes from the first set, and the other from the second set. If both sets are the same, we say the binary relation is on that set.

A binary relation may possess several properties which define its characteristics more precisely. For example, it can be reflexive (every element is related to itself), symmetric (if an element is related to another, then that element is related in the same way to the first one), and transitive—our main focus here.

Relation properties include characteristics such as reflexivity, symmetry, and transitivity that help to define and classify relations. To understand when a relation is not transitive, we must first grasp what it means to be transitive—as explained in the given exercise. A relation is transitive if, whenever one element is related to a second, and the second is related to a third, then the first must be related to the third.

Thus, the absence of transitivity can be identified when there's a break in this linkage. Given a relation R on a set A, we look for a chain of related elements and ensure that the relation completes the chain. If we find that the relation fails to complete the chain for even one set of elements, we have discovered a case where the relation is not transitive, which is critical for comprehending more complex topics in set theory and binary relations.