17.1.1. 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. This means that for any element 'a' in the set 'A', the pair (a, a) does not exist in the relation. Essentially, if you think of a relation as a way to connect elements in a set, an irreflexive relation forbids connections that loop back to the starting point.

Imagine a school where no student should be in their own buddy system; if a student can’t be paired with themselves, that’s similar to an irreflexive relation. For example, if we say students can pair for activities (like study buddies), then if student A pairs with student B, we shouldn’t allow a scenario where student A can pair with themselves.

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 we represent an irreflexive relation using a matrix, we ensure that all entries on the diagonal (which represent pairs like (a, a)) are 0. For example, if the set A has elements {1, 2}, the matrix representation will look like this:

| 0  1 |
| 0  0 |

Here, the first row and first column refer to element 1, while the second row and column refer to element 2. Since (1, 1) and (2, 2) cannot exist, their matrix entries are 0.

Think of a friendship chart among your friends where a self-friendship (friend with themselves) isn’t allowed, which is similar to the irreflexive relation. In the matrix representation, each friend cannot point to themselves, hence showing 0 for self-pairings.

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.

It's important to note that while reflexive and irreflexive relations are opposite by definition, an interesting case arises when we consider the empty set. If set A is empty, then the only relation possible is the empty relation. This means that since there are no elements to discuss, it vacuously satisfies both conditions of being reflexive (as there are no elements that can be found that do not relate to themselves) and irreflexive. Thus, an empty set can be seen as both.

Consider an imaginary scenario with a club that has no members; since there are no members, there are also no self-relationships, and thus every condition holds true by default. This is akin to saying that if no friends exist, then all friendship conditions are satisfied by default.

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

In the case where set A contains elements, it is impossible to have a relation that is both reflexive and irreflexive. If a relation is reflexive, it means that every element must relate to itself, meaning at least one pair (a, a) must exist. In contrast, an irreflexive relation requires no element to relate to itself, which contradicts the conditions of reflexivity.

Think of a standard set of friends where each friend should see themselves in a photo. If the photo is considered a valid reflection (reflexive), they all must point to themselves, negating the idea of being irreflexive.

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