[MUSIC] When we think about concepts in our everyday life, we know that some concepts can be more general than others. For example, the concept of mammals is more general than the concept of dogs, because every dog is a mammal. So dog is a subconcept of mammal, and mammal is a superconcept of dog. Let's see how we can transfer this to formal concepts. So assume that we have two formal concepts, (A1, B1) and (A2, B2). So we have (A1, B1) and (A2, B2), and they're two formal concepts of the same formal context, let’s say, (G, M, I). We say that (A1, B1) is a subconcept of (A2, B2) if A1 is a subset of A2. And that's a natural definition. So, if (A1, B1) is the concept of dogs and (A2, B2) is the concept of mammals, then, by saying that the first is a subconcept of the second, what we mean is that all dogs are mammals. Every dog is a mammal, all dogs are mammals, all dogs form a subset of all mammals. And we use the following notation to denote that (A1, B1) is a subconcept of (A2, B2). We write (A1, B1) is less than or equal to (A2, B2), so we use this sign. So this is a natural definition, but it looks a little bit strange, because we completely ignore concept intents. In this definition, we look only at A1, A2, but we ignore B1 and B2. So is there any condition that should be put on these two sets, on the concept intents? Well, let's check. So suppose that we have two formal concepts, (A1, B1), (A2, B2), and the first one is indeed a subconcept of the second one. So, by definition, we have that A1 is a subset of A2. If so, then let’s check B1 and B2. Now, B1 equals A1', because (A1, B1) is a formal concept. So, by definition of the formal concept, B1 is A1'. And A1' is the set of all attributes shared by all objects from A1. Similarly, B2 equals A2', because (A2, B2) is a formal concept, and A2' is the set of all attributes shared by all objects from A2. Now, let's look at one attribute from this set, from B2. So this attribute is shared by all objects from A2. But the set A1 is a subset of A2, this means that this attribute is also shared by all objects from A1, but then this attribute must also be included in the first set, which is the set of all attributes shared by A1 or, in other words, B1. We have just proved that every attribute from B2 is part of B1, and therefore B2 is a subset of B1. So, if A1 is a subset of A2, then B2 is a subset of B1. Now let's check if the reverse is also true. Suppose that B2 is a subset of B1. And let's look at A1 and A2. Well, A1 is B1', because (A1, B1) is a formal concept, and B1' is the set of all objects that have all attributes from B1. And, similarly, A2 equals B2', which is the set of all objects that have all attributes from B2. Now, let's look at one object from A1. This object has all attributes from B1. But since B1 is a superset of B2, this object must also have all attributes from B2 and therefore it must be part of this second set, of A2. So we’ve just proved that A1 is a subset of A2. And therefore, this relation holds in both directions. If B2 a subset of B1, then A1 is a subset of A2. And therefore, we can reformulate this definition, and we can say that (A1, B1) is a subconcept of (A2, B2) if B2 is a subset of B1. These definitions look different, but they are equivalent to each other. All right, so we've defined some binary relation, “being a subconcept of”, on our formal concepts. Let's look at some properties of this relation. Well, the first property is reflexivity. If we take a formal concept (A,B), then, by definition, it's less than or equal to, or it's a subconcept of, itself. (A,B) is a subconcept of (A,B). And indeed, this is because A is a subset of A. This property is called reflexivity. The second property is antisymmetry. If (A1, B1) is a subconcept of (A2, B2) and, at the same time, (A2, B2) is a subconcept of (A1, B1)… Well, in this case, (A1, B1) is just the same as (A2, B2), they are equal. It's the same formal concept. Well, why so? Well, because, if A1 is a subset of A2 and A2 is a subset of A1, then A1 and A2 are equal. On the other hand, if B2 is a subset of B1 and B1 is a subset of B2, then again, B2 and B1 are equal, and therefore (A1, B1) equals (A2, B2). The third property is transitivity. If (A1, B1) is a subconcept of (A2, B2), which is a subconcept of (A3, B3), in this case, (A1, B1) must be a subconcept of (A3, B3). Well, indeed, if A1 is a subset of A2, which is a subset of A3, then A1 is a subset of A3. Now, every binary relation that satisfies these three properties, reflexivity, antisymmetry, and transitivity, is called a partial order. And so we've just proved that formal concepts are partially ordered with respect to this relation. The relation of being a subconcept. [SOUND]