17.2.3. Reflexive vs. Symmetric 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. Or the element should not be related to itself. 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, that means the diagonal entry will be just consisting of 0’s or equivalently in terms of the graph representation no self loops will be present, because a will not have any directed edge to itself.

An irreflexive relation is a kind of relation where no element in the set relates to itself. For instance, if we have a set A = {1, 2}, for a relation to be irreflexive, we must not have pairs like (1, 1) and (2, 2). If such pairs are absent, the relation is marked with zeros on the diagonal entries of its corresponding matrix representation. For example, the entries (1, 1) and (2, 2) in this matrix must equal 0.

Think of this like a classroom where each student cannot be their own partner for a group project. So if we have students 1 and 2, student 1 could partner with student 2, but they cannot work in a pair with themselves. Thus, the partnerships can be illustrated as connections in a diagram showing that nobody is connected to themselves.

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. That is the only relation possible over an empty set A then this relation R is both reflexive as well as irreflexive.

A reflexive relation means that every element relates to itself, but it might seem strange to find a relation that is both reflexive and irreflexive simultaneously. However, when the set is empty, this is possible because there are no elements to negate the definitions; hence the empty relation meets the requirements for both reflexivity and irreflexivity since there are no elements to violate these conditions.

Imagine a situation where you have an empty jar: without any objects inside, you can't say that there's anything to which the jar is 'reflexively' linked and simultaneously there are also no objects to establish that it is 'irreflexively' linked. So the empty jar serves as both reflexively and irreflexively valid at the same time.

Now let us define symmetric relations, so this relation can be defined from a set A to B where B is might be different from A. So, the relation is from A to B and we say it is symmetric, so as the name suggests symmetric we want here the following to hold, whenever a is related to b as per the relation R, we need that b also should be related to a and that is why the term symmetric here.

A symmetric relation means if an element a is related to an element b, then b must also be related back to a. This symmetry must hold for all instances within the relation. For example, if we have a relationship like 'is a friend of,' once A is a friend of B, then B must be a friend of A.

Consider friendships; if you call someone a friend, that person should reciprocate that friendship back to you. So, if you are friends with someone (A is related to B), it implies they should also consider you a friend (B is related to A) for the relationship to be symmetric.

So, it is easy to see that the matrix for a symmetric relation will always be a symmetric matrix, because if you have a R b, that means the i, jth entry will be 1 and since my relation is symmetric, that means I will also have (b, a) to be present. That means if I take the transpose of M, then in the jth row and ith column, the entry will be 1.

In the matrix representation of a symmetric relation, the entries reflect this symmetry: if (a, b) is present, then (b, a) must also be present, making the matrix symmetric across its diagonal. This means that transpose (flipping the matrix) will yield the same matrix, confirming the symmetry.

Think of this like a digital matchmaking service. If person A shows interest in person B (represented by pairing in the matrix), person B should also show interest back to A. Therefore, if there is an entry showing A interested in B, there must equally be an entry showing B interested in A, maintaining a balanced connection.

But this is not symmetric because you have (1, 2) present in the relation, but you do not have (2, 1) present in the relation.

Even if a relation is reflexive (having pairs like (1, 1) and (2, 2)), it does not automatically make it symmetric. For instance, if there is a pair indicating 1 relates to 2 ()1, 2) but not the reverse (2, 1), the relation is not symmetric despite being reflexive.

Consider a situation where one person invites another to a party but does not reciprocate the invitation; the first person may have an 'invitation' to the second, yet the second does not 'invite' the first back, showcasing the absence of symmetry.

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