[MUSIC] A function is a set of ordered pairs of elements. Being ordered describes the main property which characterizes these pairs. It means that, any x value, the term means a unique value of y. In order for us to understand the mechanism underlying function, it is worth beginning from the concept of set. A set is simply an array of distinct objects which may be a group of numbers or different items and so forth. The objects which compose a set are the elements of the set. Two alternative ways of writing a set are, first of all by enumeration, and secondly, by description. If we let S represent the set of three numbers, 5, 6 and 7. We can write by enumerating these sets elements in braces rather than parentheses like S = 5, 6, 7. But, what if we speak about objects other than numbers. In particular, we can speak of sets of ordered pairs. In the concept of ordered pairs, we introduce us to the important concepts of relations and functions. If a, b is equal to b, a, then when considering the set a, b, we do not need to take care about the order of the two elements of the set. In this case, we speak about a pair of elements which is unordered. However if the order is important, that is, if this order is significant, then the set a, b, is different from the set b, a. This means that a and b, and b and a are two different ordered pairs. We can have sets of more than two elements, such as triples and so forth. In case of ordered pairs, they are called ordered sets. They are enclosed with parenthesis, rather than brackets. Ordered sets can belong to a set. If we consider a Cartesian plan, the x axis and the y axis cross each other and divide the plane into four quadrants. In this xy plane, we can individual infinite set of points. This represent an ordered pair. The first element is an x value, and the second element is a y value. If we plot the point identified like (4, 2), we will have a point different the a point identified like (2, 4). And we can now easily understand why the order is so important. There exists a generation process of the ordered pairs, suppose that we have 2 sets. x = 1, 2, and y = 3, 4. We want to find all the possible ordered pairs. And we do that by taking the first element from the set x, and the second element from the set y. The result would be what we call the Cartesian product. Based on this equation, we have the set of four ordered pairs, such as (1, 3), (1, 4), (2, 3) and (2, 4). It is also called direct product, and it is indicated by x cross y. [MUSIC]