17.5.1. Definition and Characteristics

Now let us define another special relation defined from the set to itself which is called the irreflexive relation. And the requirement here is that you need that no element should be related to itself in the relation that means you take any element a from the set A, so this universal quantification over the domain is the set A. You take every element a from the domain or the set A, (a,a) should not be present in the relation.

An irreflexive relation is a specific type of relation where no element in the set is related to itself. For any element 'a' in the set A, the pair (a, a) must not be in the relation. This implies that for any element chosen from set A, it cannot loop back to itself.

Think of it like a social network where people must not create connections with themselves. Each person can connect with others, but they cannot 'friend' themselves. This maintains a clear distinction of relationships within the group.

So, it is easy to see that if your relation R is irreflexive, then none of the diagonal entries should be 1 in the relation. So, the matrix for your irreflexive relation will be an n x n matrix. Because the relation is defined from the set A to itself and (a , a ) is not there in the relation, that means the entry number (1, 1) in the matrix will be 0.

In the matrix representation of an irreflexive relation, the diagonal elements, which represent (a, a) for each element 'a', must be 0. This means that if you visualize the relation as a matrix, every entry along the diagonal will be zero, indicating that no element relates to itself.

If we think of a classroom as our set A, the matrix representation would be like a 'no self-praise' rule. For instance, if students were to evaluate their own performance, those evaluations would not be reflected in their own scores, leading to a score of zero in that diagonal position.

So, again, let me demonstrate irreflexive relations here, so my set A is {1, 2} and I have taken the same 4 relations here. It turns out that relation R is not irreflexive because you have both (1, 1) and (2, 2) present. Similarly R is not irreflexive, R is also not irreflexive because you have (1, 1) present here, whereas R is a valid irreflexive relation because no element of the form (a, a) is present in R.

Upon examining specific relations among the elements of the set A = {1, 2}, we can determine whether they are irreflexive. A relation is classified as irreflexive if it lacks both (1,1) and (2,2). In this case, the presence of (1,1) or (2,2) in any relation disqualifies it from being irreflexive.

Imagine a scenario where judges cannot score their own performances in a competition. If a judge gives themselves any points (i.e., (1,1)), it disrupts the fairness of the evaluations. Thus, the relation would not be considered fair or irreflexive.

Now it might look that any relation which is reflexive cannot be irreflexive or vice versa... the only relation possible over an empty set A then this relation R is both reflexive as well as irreflexive.

This section clarifies that reflexive and irreflexive relations cannot exist simultaneously for any non-empty set. However, in the special case of an empty set, a relation can be both reflexive and irreflexive because there are no elements to contradict either property.

Think of a deserted island where nobody exists. Here, you can claim that each resident (if there were any) knows themselves (reflexive) and simultaneously, there are no residents to relate to (irreflexive). The paradox of empty relationships illustrates both properties perfectly without contradiction.

--