21.4.6. Part F: Neither Reflexive nor Irreflexive Relations

The last part of the question, I have to find out the number of relations, which are neither reflexive nor irreflexive. So, not reflexive means this universal quantification should be false that means at least 1 element a, should be there in the set S such that (a, a) is not present in the relation, then only this universal quantification can become false. And, not irreflexive means, this second universal quantification is false, that means you have at least one element a in the set S such that (a, a) is present in your relation then only this second universal quantification can be false.

To understand relations that are neither reflexive nor irreflexive, we need to define what it means for a relation to be reflexive or irreflexive. A reflexive relation includes all pairs of the form (a, a) for each element a in the set. In contrast, an irreflexive relation excludes all such pairs. Therefore, to be neither reflexive nor irreflexive, we require a relation that includes at least one pair (a, a) and also excludes at least one pair (a, a). Thus, we cannot have a relation that is simultaneously reflexive (which requires all pairs (a, a)) and irreflexive (which requires no pairs of this form).

Imagine a classroom where a teacher decides that in a group project, everyone must either work together with every member (reflexive) or not work with anyone (irreflexive). But to find a scenario that's neither, think of a group where some students chose to work together while others did not, signifying that not everyone is included, but also that some friendships (or working relationships) do exist.

So, if I consider this n2 ordered pairs, then what I can say here is that, since my set S is non empty, so I stress I assumed here n is greater than equal to 1. So, since my set S is non empty, I cannot have a relation which is simultaneously reflexive as well as irreflexive, right?

Given that our set S has at least one element (n >= 1), it is impossible for any relation on S to meet the conditions of being both reflexive and irreflexive together. Since reflexive relations demand inclusion of all (a, a) pairs, yet irreflexive demands their complete exclusion, these two properties contradict each other. Thus, if a relation were reflexive, it could not also be termed irreflexive.

Think of a sports team required to follow set rules: everyone must show up for training (reflexive), while at the same time, nobody is allowed to be present (irreflexive). This is like saying an athlete must be both totally committed and entirely absent at the same time, which is logically impossible.

So, I have to exclude all the elements along the diagonal. It turns out that if I try to find the number of reflexive and irreflexive relation simultaneously with respect to the options that I have for the non diagonal elements then the counting might become slightly tricky. So, instead what I do here is, I find out the number of relations which are either reflexive or irreflexive and subtract it from the total number of relations, which I can form over the set S.

To count the number of relations that are neither reflexive nor irreflexive, the approach involves first determining all potential relations that could be formed from the set S. We know that the total number of relations over a set with n elements is 2^(n^2). Next, separately calculate the number of reflexive and irreflexive relations. By subtracting the count of either type of relation from the total number, we can accurately identify how many relations fall into the category of neither.

Imagine a store with various products (elements of the set). You have rules for each product: some products must always be displayed (reflexive), and others must never be displayed (irreflexive). To find products that are neither displayed nor should never be displayed, you take all products and remove those that fit into the 'always displayed' and 'never displayed' categories. What remains is your target group.

--