17.3.1. 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...

An irreflexive relation is a specific type of relation where no element in the set relates to itself. For example, if A is a set containing elements like {1, 2}, then for an irreflexive relation R, you cannot find pairs (1, 1) or (2, 2) in R. This concept is important because it deals with the diagonal entries in a relation matrix representing this relation. In the matrix, each diagonal entry corresponds to an element relating to itself, which in irreflexive relations must always be 0.

Think of irreflexive relations like a classroom where no student gives feedback to themselves after a class discussion. Instead, every feedback is given from one student to another student. In this scenario, you never see (Student A, Student A) in the feedback forms, just like in an irreflexive relation you never see (a, a).

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...

In matrix representation of relations, an irreflexive relation will be depicted with a matrix that has all its diagonal elements as 0. For example, if A = {1, 2}, the matrix form of an irreflexive relation will look like this: [0 1; 1 0] or [0 0; 0 0] where there are no instances of (1, 1) or (2, 2). This visual representation helps in easily spotting the nature of the relation.

Imagine a friendship network where every friend is connected to others but not to themselves. A matrix could represent who is friends with whom, where the diagonal entries (representing self-friendships) are always zero, indicating that no one is friends with themselves.

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...

To determine if a relation is irreflexive, we can examine various pairs. If any relation contains pairs like (1, 1) or (2, 2), it cannot be irreflexive. In the example with set A = {1, 2}, if one relation contains both (1, 1) and (2, 2), it is not irreflexive. However, if a relation includes pairs like (1, 2) and (2, 1) without (1, 1) or (2, 2), it is indeed irreflexive.

Consider a group of two people discussing their favorite books. If one person claims, 'I love my own book,' the relation between them becomes reflexive. But if both solely discuss each other's favorites, like in a critique where no one talks about their own work, it mirrors an irreflexive relation.

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...

In certain cases, particularly when discussing the empty set, a relation can be both reflexive and irreflexive. For the empty set, no elements exist to violate the conditions of these relations. Therefore, an empty relation can vacuously satisfy both attributes because there are no instances to break the rules.

Imagine an empty classroom: there are no students, so no one interacts with themselves (irreflexive) and simultaneously, because there are no students, it also upholds the notion of reflexivity because there’s nothing to contradict it.

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

When the set A contains one or more elements, it is impossible for a relation to simultaneously be reflexive and irreflexive. This is due to the definitions where reflexive implies the presence of (a, a) and irreflexive implies the absence of (a, a). Thus, both cannot coexist when real elements are considered.

Think of a group of friends being known to each other. Each person recognizes themselves (reflexivity), but if they are only considerate towards others and never refer to themselves, it can create a paradox. In this scenario, you cannot both acknowledge oneself and not acknowledge oneself simultaneously.

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