18.7.1. Reflexive Closure

Next let us define what we call as closure of a relation. So, what do we mean by this? So, imagine you are given a set A that may be finite or infinite and you are given a relation R over the set A that means a relation R is a subset of A x A and I have some abstract property P, it is some abstract property and I am interested to check whether the relation R over the set A satisfies this property P or not? If it satisfies the property well and good. But, that may not be the case R that is given to you need not satisfy the property. So, the closure of the relation R is defined with respect to this property P. If you change the property P, the closure will change. So, what is that is the closure of the relation R with the property P? Well, it is the smallest superset of R which has the property P. Pictorially what I am trying to do here is if my relation R already satisfies the property P, I do not need to add anything to the relation R. I do not need to actually add any extra element to the relation R to satisfy the property P. But if my relation R does not satisfy the property P then I will be interested to introduce new ordered pairs in the relation R and convert it into another relation S, so that the expanded relation S satisfies the property P that is what I am trying to do here, this S you can imagine as an expanded version of R and this S is also going to be a relation over the set A itself. I am including the original relation R that is carried as it is. On top of that I am adding or I may add few extra elements and try to ensure that expanded R which is S satisfies the property P, but I am not going to do the expansion arbitrarily; I am interested in the least possible expansion, least expanded version, what I mean by least? That means this is the minimal expansion which I need to do in order to ensure that the relation S satisfies the property P that is important. Otherwise, what is a big deal in expanding the relation R? You keep on adding any arbitrary number of elements definitely you will more or less soon get an expanded version which will satisfy the property P. So, we are interested in the smallest possible expansion.

The closure of a relation describes how we can enhance a given set and its relationship under certain properties. When provided with a set A and a relation R that might not fulfill a specified property P, the goal is to find the smallest extension of R that meets this property. For instance, if R doesn't have this property, you would add the minimal set of ordered pairs needed to comply with P. If R already satisfies P, we don’t need to alter it at all. Therefore, the closure is about creating the least expanded version of R while ensuring the new relation still adheres to property P.

Think of a flower garden that you're trying to make more beautiful. If some flowers already match your vision (let's say they're all blue), you don’t need to change anything. But if you realize some flowers are red and you want blue ones, you just need to replace those as few as possible to keep the overall appearance cohesive. Similarly, in relational algebra, when a relation lacks a property, we introduce the necessary elements to fulfill that property without unnecessary additions.

So, let us see some examples of this abstract property P and how the resulting closure looks like. So, my first abstract property is the reflexive property and this gives us what we call as reflexive closure of a relation. So, you are given a relation R over the set A and I am interested to see whether this relation R satisfies the reflexive property or not, reflexive over the set A, so that is what is the reflexive closure of the relation R. So, how can you construct a reflexive closure of R? Well you just take the union of R with all ordered pairs of the form (a, a) where a is present in your set A. If your, (a, a) is already there in the relation R then as per the union definition, you will not be including it again. Remember, union means if (a, a) is present in R as well as in this new relation, so this new relation I am calling it as Δ relation. So, this Δ relation you can imagine it is consisting of all ordered pairs of the form (a, a ) such that a is present in A. So, if (a, a) is already there in R, it will not be included again but if (a, a) is not present in R then due to the union, due to taking union with this Δ, it will be now added to the relation R and now you can see that this is your expanded R that may be same as R itself, in case if your relation R is already reflexive then you are not going to add any extra elements. So, this expanded R will have the original elements of the relation R plus this expanded R will satisfy the reflexive property.

The reflexive closure involves modifying a relation so that it meets the reflexive property, which states that every element must relate to itself. To create the reflexive closure of a relation R on a set A, you combine R with all pairs of the form (a, a) for every element 'a' in A. This means if the pair (a, a) is not already there in R, you'll add it through a union, ensuring every element of A is included in a way that fulfills the reflexive requirement. If (a, a) is already in R, it remains unchanged. This process ensures that R evolves into a new relation that meets the reflexive property.

Imagine a classroom where each student must state their name (attend class) to be counted present. If all students are already saying their names, then attendance is fully met (the relation is reflexive). However, if a couple of students forgot to announce their presence, you need to add those names to the attendance list. By combining those names with the already announced ones, you create a complete attendance list, ensuring everyone is accounted for, just like ensuring every element relates to itself in the reflexive closure.

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