17.1.2. Examples of 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 specific type of relation where no element of the set relates to itself. For instance, if we consider a set A and an element a within that set, the condition states that the pair (a, a) cannot exist in the relation. This means that for every element in the set, it cannot be associated back to itself.

Imagine a classroom where students can only interact with their peers and not with themselves. If we think of the students as elements, then an irreflexive relation says that no student can mark themselves for a group project; they can only mark others.

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.

When representing an irreflexive relation using a matrix, if the relation is defined over set A, the matrix will not have any 1s on the diagonal. This is because the diagonal represents pairs (a, a), and since these cannot exist in an irreflexive relation, all diagonal entries are set to 0. Thus, the entries (1, 1), (2, 2), etc., will all be zeroes.

Think of a sports team where players can only play with others and never against themselves. If we make a matrix where rows and columns represent players, a zero on the diagonal indicates that Player 1 cannot play against Player 1, which is a self-match.

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 might seem that relations cannot be both reflexive and irreflexive, yet there is an exception when the set A is empty. In this case, there are no elements to relate back to themselves, so the conditions for both properties hold vacuously. Therefore, the empty relation on an empty set is considered both reflexive and irreflexive because there are no pairs (a, a) to contradict either property.

Imagine an empty classroom with no students. In this scenario, since there are no students, there's no way for any student to relate to themselves, making it both reflexive and irreflexive by default, just like how an empty relation does not contradict any rules.

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

However, if the set A contains at least one element, then it becomes impossible for a relation to be both reflexive and irreflexive simultaneously. A reflexive relation requires that each element relates to itself, which contradicts the definition of an irreflexive relation where no element can relate to itself.

Consider a basketball team that must have at least one player. In this team, if we state that every player can only play with others and never with themselves, we cannot also say that they must have the ability to play with themselves; it's either one or the other, similar to the relationship rules.

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