Can you imagine a telephone book which is not ordered? The theme of this lecture is the relations of order, or ordering relations. They allow to compare elements with respect to some features and, in consequence, to arrange elements in a particular order which simplifies the process of search superbly.

In many situations we are to execute the operation of searching in a set. It is required in science and in applications. Suppose that a searched set is represented by a list of randomly puted objects. Then the algorithm of searching need to go sometimes trough the whole list and is slow. If however the succession of elements is given by an ordering relation known to us, then the goal may be reached very quickly.

The reader probably knows the game of "20 questions". The player A has chosen a natural number n less than 1000000. The player B is to guess the number n. The player B can formulate questions in such a way that player A answers either YES or NO. The player B can ask at most 20 questions. Now, we observe that the player B has a winning strategy because the set of natural numbers is a linearly ordered set and so is the subset of natural numbers less than one million. He ought to pose the questions in a way that allow to fixe in which half of the current interval (at the beginning 1-1000000) is the searched number.

Question: Is it important that we limited the magnitude of the chosen number? Can we allow the player A to choose an arbitrarily big natural number without changing another parameters of the game?

It is impossible to over estimate the significance of ordering relations. Every data base, every dictionary, every catalogue uses an ordering relation. Our everydays activities would be much less efficient without orderings.

Apart from ordering relation, this lecture presents the notions of maximal and minimal elements, as well as the least and the greatest elements. Moreover, we shall introduce the notions of bounds: lower and upper bounds of a subset of a given set. This will lead us to the notions of  the least upper bound and the greatest lower bound. To illustrate ordering relations we will use the diagrams of Hasse .
We conclude this lecture considering the problem: Is it possible to equip any set with an ordering relation? Is every set well-ordered? The answer to the first question is: yes. Consider the identitity relation i.e. the set of pairs (x, x) for all x in X . The second question has an easy answer only for the case of finite sets. Each sequence of all elements of the given finite set defines a well-ordering in it.
An interesting fact that is beyond the scope of this couse is that for infinite sets the answer is much more complicated. The mathematicians have no common answer to it. Some variants of the set theory accept the so called axiom of choice. In these theories one can prove that every set can be viewed as a well-ordered set. In some other the answer is negative.