This lecture extends earlier lectures on binary relations. We aim to present the equivalence relations which are a special case of binary relations.  The notion of equivalence relation is an abstract notion of mathematical character, nethertheless, it has a big significance in practice. Frequently, when we are dealing with large sets, i.e. the sets where the number of elements is big, it is more convenient to group objects with respect to some features of interest and later work with the representatives of groups instead of each object separately. Each equivalence relation determines a partition of the set onto disjoint classes, so called equivalence classes or classes of abstraction, that gather all elements that are pairwise equivalent. The principle of abstraction, which is related to it, is one of the most used and pervasive tools in mathematics.

Principle of abstraction, which is also called the principle of identification of equivalent objects, is the most important theorem of this chapter. In section 5 we present the famous applications of the principle - we show how the integer numbers and rational numbers are constructed as equivalence classes. It turns that if we define an equivalence relation on the sequences of rational numbers in a proper way, then the real numbers are nothing else but abstraction classes of sequences of rational numbers. The method is known as Cantor's method of construction of real numbers.

We are calling the attention of the reader to the notion of congruence which is also introduced in the lecture. The notion of congruence is very important in informatics. For example, one programming language may have several different realizations, compilers. We expect that they all be equivalent in this sense that regardless of a chosen realization our program behaves in the same way. This property will be assured, if every operation executed on equivalent objects will give equivalent results, i.e. the equivalence must be a congruence relation. More on congruences will be told in lecture 10.