[MUSIC] Let's see what implications hold in our context of triangles. So from the previous video, we know that there are three conditions we may want to check if we want to check if an implication holds. So implication A —> B holds in the context K if every object intent is a model of A —> B; so, for all g, we have that g' is a model of A –> B; or if B is a subset of A''; or if A' is a subset of B'. So these three conditions are equivalent to each other. And if we want to check if an implication A –> B holds in a context, we can check any of them. So, let's see if the implication {a, b} –> {c} holds in this formal context. Well, let's use the first condition. So we need to check that every object intent is a model of {a, b} –> {c}. Let's look at the first object intent, {b, d}. It's a model of {a, b} –> {c}, because {b, d} doesn't contain {a, b}. The same is true for the second object intent, {b, e}: {a, b} is not part of {b, e}; and for the third object intent, {c}: {a, b} is not a subset of {c}. For the fourth object, the case is different. Here {a, b} is indeed a subset of {a, b, c}. And for this object intent to be a model of {a, b} –> {c}, it must contain {c}. And it does: {a, b, c} contains {c}. So the fourth object intent is also a model of {a, b} –> {c}. And so are the fifth, the sixth, and the seventh, because neither of these three objects contains {a, b}. So neither of them violates this implication, {a, b} —> {c}. {a, b} implies {c} is indeed a valid implication of our formal context. Let's see what it tells us. a is equilateral, b is isosceles, and c is acute-angled. So it says that every equilateral, isosceles triangle is also acute-angled, and this is true. Because if a triangle is equilateral, then all its angles are 60 degrees, which is less than 90 degrees, so they are acute. And notice that we don't even need this condition, isosceles. Well, every equilateral triangle is isosceles. But it's sufficient for a triangle to be equilateral, to say that it's acute-angled. So it looks like we can make this implication sound stronger. We should be able to say that {a} implies {c}. Let's check if this implication indeed holds in our formal context. But we'll use the second condition for a change. So we need to check that {c} is a subset of {a}''. So what's {a}''? {a}'' is the set of all attributes shared by all objects that have attribute a. There's only one object that has attribute a. It's T4. And it also has b and c. So {a}'' is {a, b, c}, and indeed {c} is a subset of {a, b, c}. So this implication, indeed, holds in our formal context. Let's make this implication stronger in a different way. Let's remove a from the premise and check whether it holds in the context. So does {b} imply {c}? Let's use the third condition, A' is a subset of B'. So let's check this. So we need to compute {b}'. {b}' is the set of all objects that have attribute b. An that's T1, T2, T4, and T6. And now we have to compute {c}'. {c}' is the set of objects that have attribute c. So it's T3, T4, and T6. This is {c}'. And this is not a subset of this. So {b}' is not a subset of {c}'. So this implication doesn't hold. Well, why doesn't it hold? This set contains T1 and T2, that are not part of this set. So these two triangles, T1 and T2, are actually counterexamples to this implication. They have attribute b, but they don't have attribute c. Let's look at them. Well, {b} –> {c}, what this implication is all about? It tells us that every isosceles triangle must be acute-angled. So, T1 is an obtuse-angled isosceles triangle. It has two equal sides, but it has an obtuse angle. So it's not acute-angled. And T4, sorry T2, is a right-angled isosceles triangle, which is also not acute-angled. So these are two counterexamples to this implication. And in general, this is how we can get counterexamples for an implication A –> B, if there are any. The counterexamples are precisely objects that belong to A', but don't belong to B'. [MUSIC]