[MUSIC] Now let's look at the formal definitions of context and concepts. A format context is very simple entity: it's just two sets and the binary relation between them. We treat one set, G, as a set of objects and the other set, M, as a set of attributes. Objects are described in terms of attributes, and this is the role of the binary relation I: it specifies which objects of G have which attributes of M. Formally, I is a set of pairs, where the first element is an object from G and the second element is an attribute from M, and this object has this attribute. This is an example of a formal context represented by a so-called cross-table. It has seven objects, and five attributes. Objects are triangles, and they correspond to rows in this table. Attributes are shown in columns, and they are various properties of triangles. A triangle is equilateral if all its sides are equal, and isosceles if at least two sides are equal. In acute-angled triangles all angles are less than 90 degrees, while an obtuse-angled triangle has a greater angle, and the right-angled triangle has an angle of exactly 90 degrees. A cross in a table cell indicates that the object corresponding to the row of this cell has the attribute corresponding to its column. Thus, T1, as an isosceles acute-angled triangle, has crosses only in cells b and d, and only right-angled triangles T2 and T7 have crosses in the last column. Given a formal context, we define two derivation operators. One acts on objects, the other on attributes, but, more often than not, they're denoted by the same symbol. It can be the name of the incidence relation of the context, or, when it is clear what context and what relation is meant, these operators are denoted by primes. So, if A is a set of objects, then A' is the set of all attributes from the context shared by all objects from A. If B is a set of attributes, then B' is the set of all objects from the context that have all attributes from B. Let's look at some examples. Take an object set consisting of the second triangle, T2. {T2}’ is the set of all attributes of T2, that is, the set {b, e}, {isosceles, right-angled}. This is called the object intent of T2, and we often write object intents without curly brackets. So, an object intent is the set of all attributes this object has. Let's add T4 to our object subset. {T2, T4}' is the set of all attributes common to both objects in this set. There is only one such attribute, b; so, {T2, T4}' equals the single element set, {b}. But what is {b}’? It must include all objects that have attribute b, and we have two such objects in addition to the second and the fourth triangles, so, four objects in total. They form the attribute extent of b. An attribute extent is the set of all objects that have a particular attribute, and again, we may meet curly brackets around the attribute in our notation for attribute extents. Another example. Since the second and the third triangles have no common attributes, {T2, T3}' is the empty set. What is the {}'? Well, this question is a bit ambiguous, because, remember, we have two prime operators: one is applied to object subsets, and the other to attribute subsets. Up to now, there was no need to specify which prime operator is meant, because this was clear from how we used it. When we use prime in conjunction with an object subset, it's the operator that maps object subsets to attribute subsets. And if we use prime in conjunction with an attribute subset, it's the other of the two operators. But the empty set is a subset of both G and M, so, it can be treated as either an object subset or an attribute subset. Hence, we should say explicitly what prime operator we have in mind, when we apply it to the empty set. It may be necessary to be explicit in some other cases, too, for example, when the sets G and M are the same or, at least, overlapping. Back to our empty set, if we treat it as an attribute subset and apply the second of the two derivation operators, then the {}' is G, the set of all objects. And it's like this for any formal context. Indeed, for an object to belong to B’, where B is an attribute set, this object must have all attributes from B. But if B is empty, this condition is trivially satisfied by all objects, and so they all belong to B'. Similarly, if we apply the first derivation operator to the empty set, we will obtain the set of all attributes. So, it's important to remember, that, even if we use the same notation for the two derivation operators, they are two different functions. Also, it is sometimes necessary to be able to talk about several formal contexts and their derivation operators. In this case, the prime notation may become ambiguous and we will usually denote the derivation operators by the corresponding incidence relation symbols. But for now (and for most of this course), we'll stick to primes. So, let's look at some of their properties. Let A and C be object subsets. Three simple properties. The first one: if A is a subset of C, then C' is a subset of A'. Why should this be true? Well, C' is the set of all attributes shared by all objects from C, which include all objects from A, because A is a subset of C. Hence, these attributes from C' are shared by all objects from A, and therefore, they all belong to A'. Thus, C' is indeed a subset of A'. I will leave the proofs of the second and the third properties as exercises. The second property says that A is always a subset of A''. And the third property essentially says that you shouldn't apply prime operators more than twice in a row if you hope to get anything new, because A' equals A'''. Now that the second prime in the last property is different from the first prime, A is a set of objects, but A' is a set of attributes; so, we apply the first derivation operator to A, and the second one to A’— and then again the first one to A’, which is a set of objects. And of course, we have similar three properties relative to attribute subsets. Here they are. These properties show how the two derivation operators interact. And those who are familiar with this terminology might have noticed that the two operators constitute a Galois connection between the powersets of G and M. [SOUND] [MUSIC]