17.1. Irreflexive Relation

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 condition applied to elements of a set whereby no element can be related to itself. For every element 'a' in a set 'A', the pair '(a, a)' is not part of the relation. This means that when we think of the relationship between elements, no element can have a self-connection or self-loop.

Imagine a social network where each user can connect with others; an irreflexive relation would be like saying that no user can send a friend request to themselves. Therefore, in this social network, even if you can send friend requests to many people, 'John cannot be friends with John.'

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, that means the entry number (2, 2) in your matrix will be 0 and so on.

In a matrix representation of an irreflexive relation, the diagonal elements (which correspond to pairs of the form '(a, a)') will always be zero. This is because an irreflexive relation prohibits any element from relating to itself. As a result, the matrix will have zeros along the diagonal, indicating that no self-relations exist.

Think of a classroom seating arrangement. If we use a matrix to represent who sits next to whom and no student can sit next to themselves, the positions along the diagonal (1,1), (2,2), etc., representing 'student 1 next to student 1', will be marked as 0 to show that they can't sit next to themselves.

In terms of the graph representation, no self loops will be present, because a will not have any directed edge to itself, and so on.

When depicting an irreflexive relation using a directed graph, there will be no self-loops at any node. A self-loop is a connection that points back to the same node, representing a relationship of an element with itself. Since by definition, an irreflexive relation prohibits self-relations, nodes in the graph will not have edges leading back to themselves.

Consider a network of roads between cities. If there are no roads that allow a city to connect to itself, it means that you can't drive in a loop from City A back to City A. Just like in our directed graph, you won't see any paths that loop back.

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, whereas R is a valid irreflexive relation because no element of the form (a, a) is present in R.

In this example, set A consists of the elements {1, 2}. We check different relations to see if they are irreflexive. If any relation contains the pairs (1, 1) or (2, 2), it fails to be irreflexive. On the other hand, if none of the self-relation pairs are present, the relation is successfully irreflexive.

Picture a sports league where each team can compete with any other team but never with themselves. If you record the matches in pairs (Team1, Team1), (Team2, Team2), the league wouldn't be irreflexive since each team is trying to 'play against' themselves, which isn't allowed.

Now it might look that any relation which is reflexive cannot be irreflexive or vice versa. However, 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.

While reflexive and irreflexive seem mutually exclusive, a special case arises with an empty set. If our set A is empty, then the only possible relation is the empty relation, which satisfies both definitions vacuously because there are simply no elements to contradict either definition.

Think about a group of people planning a gathering. If there are no people (an empty set), then you cannot have anyone relate to anyone else, be it reflexively or irreflexively. It’s like saying an empty box can fit into any category just because it's empty, meaning it doesn't violate any rules.

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 elements, a relation cannot satisfy both reflexive and irreflexive properties simultaneously. If any element in A has a self-relation, it cannot be irreflexive, and if all pairs (a, a) are absent, then it cannot be reflexive.

Consider a social network with actual users. Each user has some connections (relationships). If a user is friends with themselves, then they cannot be in a network where no one can connect with themselves. Therefore, reflexive and irreflexive properties cannot coexist in a populated social context.

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