[SOUND] [MUSIC] Now let's define an operator. We'll call it the preclosure operator. L bullet, that takes a set to its smallest preclosed superset. Because preclosure sets form a quasi-system, such operator must be a close operator. So the preclosure operator. Is a closure operator. So once again, what's happening here? We started from a closure operator double prime. Now we know that some sets are preclosed with respect to this operator, double prime. Now we define another operator, L bullet, that takes every set through it's smallest preclosed super set. Preclosed with respect to double prime. At this rate it turns out Is also a clause operator, but it's a different clause operator. Now let's define. First let's define an intermediate operator. Assume that L is the economical basis. The economical basis of our closed operator double prime. And so for this operator, X to L bullet as follows. Well it's going to be a super set of X. So it contains X, but it also contains some other attributes. And we go through, okay, it will contain sets P double prime, such that P double prime is that conclusion of some implications from L. So we have an implication P implies P double prime in L. But also what's important is that P is a proper subset of X. So, we start with X. We'll look at implications in L. And if implications of the form P implies P double prime. And if P is a proper subset of X, then we add our P double prime, the conclusion of the implication to X. Well this is almost like comparing the closure, with respect to L, except that we're not looking at implications that have X as a premise. We're only looking at proper subsets of X. That's just one step in computing the preclosure operator. And then we're starting with X. We compute X two L bullet. We get another set. Then we're, again, apply this operator, this intermediate operator, we get another set. And we continue like this as long as we get anything new. If at some point nothing changes after the application of L bullet, then we stop. And we now know we have computed the preclosure of X. So we iterate. Until. We obtain a set. L bullet (X) which is X to L bullet do L bullet and so on. So, at some point, we obtain a set and we'll call it L bullet effects, such that any further application of this intermediate operator changes nothing. And this is the definition of our preclosure operator. It can be shown using prepositions from the previous video that this is indeed the preclosure operator. There is L bullet (X) is indeed the smallest preclosed superset of X. Note also that when we computed L bullet (X), we only needed implications whose premises were and were proper subsets of X, and then proper subsets of L bullet (X). So we'll use this property to compute the implication basis in the next video. [SOUND] [MUSIC]