17.1.3. Reflexive and Irreflexive Relations

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 relation where no element is related to itself. For example, if we have a set A, like {1, 2}, an irreflexive relation would mean that for each element a in A, the pair (a, a) is not in the relation. This means for both elements, 1 and 2, the pairs (1, 1) and (2, 2) cannot exist. Consequently, this leads us to note that an irreflexive relation contains none of those types of pairs.

Think of a group project where no team member can give feedback about their own work. Each team member can provide feedback to others, but they cannot provide feedback on their own contributions. This situation mirrors an irreflexive relation.

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. Similarly (a , a ) is not there in your relation.

In a matrix representation of an irreflexive relation, if we denote the relation by a square matrix, then the diagonal entries—representing pairs of the form (a, a)—will all be 0. This represents that no element is related to itself. For instance, in a relation where the set is {1, 2}, the matrix might look like this: [[0, 1], [0, 0]], indicating that 1 is related to 2, but neither is related to themselves.

Imagine a chess game where every piece cannot capture itself; they can only interact with other pieces. In this analogy, the matrix would show that none of the pieces (1 for pawn, 2 for knight) can capture themselves, leading to a '0' in the diagonal entries of their interaction matrix.

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 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.

By checking different relations for a set A = {1, 2}, we can see which are irreflexive. For example, if R1 includes (1, 2) and (2, 1), it is irreflexive, as neither (1, 1) nor (2, 2) are involved. Conversely, if R2 includes (1, 1) and (2, 2), it cannot be irreflexive because those pairs indicate reflexivity.

Imagine a classroom where each student gives feedback on others' presentations but does not evaluate their own. If students A and B evaluate each other's work, this reflects an irreflexive relation since they are not reviewing their work.

Now it might look that any relation which is reflexive cannot be irreflexive or vice versa but or equivalently can we say that is it possible that I have a relation which is both reflexive as well as irreflexive defined over the same set A. Well the answer is yes because if you consider the set A equal to the empty set, and if you take the relation R, which is also the empty relation.

It is indeed possible for a relation to be both reflexive and irreflexive, but this is a very particular case where the set A is empty. In this case, since there are no elements in A, there cannot be any pairs (a, a) present, satisfying irreflexivity just as well as reflexivity. This condition highlights how definitions can blur under certain circumstances.

Imagine an empty room where nobody is present. You can't have any one person feeling connected to themselves or disconnected because, quite simply, no one is there. This scenario illustrates how a relation can be both reflexive and irreflexive when no elements exist.

If A is non empty, then definitely you cannot have a relation which is both reflexive as well as irreflexive.

In a non-empty set A, such as {1, 2}, reflexive and irreflexive cannot coexist. A reflexive relation implies that each element must relate to itself (having pairs like (1, 1) and (2, 2)), while an irreflexive relation explicitly states that these self-relation pairs cannot exist. Therefore, it’s logically impossible for a non-empty set to satisfy both properties simultaneously.

Consider a soccer team where each player must pass the ball to themselves (reflexive) versus a rule that states nobody can pass the ball to themselves (irreflexive). If there are players on the field, they cannot follow both rules without contradiction.

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