17.2. Symmetric Relations

Now let us define symmetric relations. This relation can be defined from a set A to B where B 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 establishes a two-way connection between elements of two sets. If one element, say 'a', is connected to another element 'b', then 'b' must also be connected back to 'a'. This property is crucial in certain mathematical contexts, as it indicates mutual relationships.

Think of a friendship between two people. If person A is friends with person B, then it naturally implies that person B is also friends with person A. Thus, the friendship relationship is symmetric.

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 context of matrices, a symmetric relation can be represented in a matrix form where the entries reflect the relationships between elements. If the entry at position (i, j) in the matrix is 1 (indicating a relation), the entry at (j, i) must also be 1 due to the symmetric property. Thus, a symmetric matrix remains unchanged when transposed.

Imagine a friendship graph where each person is a node. If person A is friends with person B, the graph shows an edge from A to B and also from B to A. This results in a symmetric adjacency matrix where the connections are mirrored.

This does not mean that you need every element of the form (a, b) and (b, a) to be present in the relation R; this is an implication. The implication here says that if (a, b) is present in R, then only you need (b, a) to be present in R.

It's important to note that not every pair (a, b) and (b, a) must be present in a symmetric relation. The requirement is conditional: if (a, b) exists, then (b, a) must also exist. However, if (a, b) does not exist, the presence or absence of (b, a) is irrelevant for determining symmetry.

Consider pairs of shoes. If you have a left shoe (a) that matches with a right shoe (b), you also have the matching right shoe that goes with the left. However, if you only have a left shoe, it doesn't mean the right shoe must be present. The relationship is only intrinsic when both shoes are present.

Let us do this example, I have set A = {1, 2} and I am defining various binary relations from A to A itself. Now which of the following relations are symmetric? The first relation is a symmetric relation because this condition is true here. I can say that since (1, 1) is present in the relation, I also have (1, 1) present in the relation. Similarly the relation R is a symmetric relation.

In this example, the students can consider set A consisting of two elements. When defining various relations, it's crucial to identify which relationships meet the symmetry criterion. The presence of pairs like (1, 1) indicates a self-relation, which also satisfies the symmetry condition. Observing pairs helps in determining the symmetric status of given relations.

Using the same friendship analogy, if each friend considers the other as their friend (like both A and B are friends with each other), then their relationship is symmetric. For various friendships, we check if their connections remain two-way.

So, here is a question for you, can I say that every reflexive relation is also a symmetric relation? So, remember reflexive relation means every element of the form (a, a) will represent in R. And apart from that I might have something additional also present in the relation.

The statement that every reflexive relation is symmetric is not universally true. A reflexive relation ensures that every element relates to itself (like (a, a) for every 'a'). But it doesn't guarantee that distinct elements will relate symmetrically. A counterexample can easily illustrate this discrepancy.

Imagine a group of students where each student claims their own work as outstanding (this represents reflexivity). Just because each student acknowledges their own work doesn't imply they all praise each other's work equally, illustrating the differences in reflexive versus symmetric relations.

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