17.6. Summary of Binary 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 type of relationship where no element relates to itself. For example, if you have a set A containing elements, an irreflexive relation means that for every element 'a' in A, the pair (a, a) is not part of the relation. This means no element is allowed to point to itself in the relationship.

Think about a situation where parents do not refer to their own children - in this case, each parent cannot have a direct relationship with themselves (a parent referencing themselves as 'my parent' is not valid), representing an irreflexive relationship.

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.

In matrix representation of a relation, if a relation is irreflexive, all diagonal entries in the corresponding matrix will be 0. This indicates that no element is related to itself. For instance, if 'A' consists of 2 elements, the 2x2 matrix will have all zeros on the diagonal, such as: [[0, x], [y, 0]], indicating ‘x’ and ‘y’ relationships between different elements.

Imagine a friendship matrix where friends are listed by rows and columns. If nobody considers themselves a friend, the diagonal elements (which represent friendships with themselves) would all be 0, as no one has self-friendship.

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, R is also not irreflexive because you have (1, 1) present here, whereas R is a valid irreflexive relation because no element of the form (a, a) is present in R.

In the example provided, the set A consists of numbers 1 and 2. A relation is checked to see if it’s irreflexive by looking for pairs like (1,1) and (2,2). If either pair is present, then the relation fails to be irreflexive. The only time the relation is valid as irreflexive is when none of these pairs exist.

Think of a sports match where players cannot play against themselves. A list showing players' matches would be considered irreflexive if no player is listed as playing against 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 special case occurs with empty sets. An empty set inherently satisfies conditions of reflexivity and irreflexivity as there are no elements to violate either condition. Therefore, the relation defined on an empty set is both reflexive and irreflexive at the same time.

Imagine an empty classroom with no students. Since no students exist, there are no cases of students not being present to learn nor instances of being present. The absence of students means both rules—their presence and absence—are satisfied.

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.

Symmetric relations indicate a mutual relationship. If an element from set A is related to an element in set B, then the reverse must also hold; if 'a' relates to 'b', then 'b' must relate back to 'a'. It's a one-to-one requirement that must reciprocate.

Consider a friendship: if person A is friends with person B, then person B must also consider person A as their friend, indicating a symmetric relationship.

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 entrry will be 1 and since my relation is symmetric, that means I will also have (b, a) to be present.

In a symmetric relation, the matrix shows that if one position has a connection (1), then the transpose of that position will also have a connection, indicating symmetry across the diagonal. Thus, if (a, b) gives a 1 in cell (i,j), then (b, a) will also be 1 in cell (j,i), reinforcing that symmetry.

Imagine a communication network where if person A sends a message to person B, then person B should also be able to send a message back to person A, just like two-way communication represents a symmetric relationship.

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

Just because a relation is reflexive, meaning all elements relate to themselves, does not guarantee it is also symmetric. If there is a (1, 2) pair without the corresponding (2, 1), symmetry is violated. Reflexivity is not sufficient for symmetry.

Think of a person's birthday. Just because it's their birthday (reflexivity), it doesn't mean that they know it's someone else's birthday unless there's mutual awareness of both events—which reflects symmetry.

Now the next special relation is the asymmetric relation and the condition here is, if you have a related to b in the relation, then you demand that b should not be related to a.

Asymmetric relations require a one-way connection: if 'a' relates to 'b', then 'b' cannot relate back to 'a'. This means that such relations can never have a two-way relationship; if one exists in a specific direction, the other must not.

Think of a one-way street: if you can drive from point A to point B, you cannot come back from B to A. Thus, the relationship is asymmetric, just like asymmetry in directional movement.

You take any i, jth entry, ith row and jth column, you can have at most one of the entries i, j or j, i being 1 in the matrix.

In the matrix form, for an asymmetric relation, you can't have both (a, b) and (b, a) be 1 at the same time. If 'a' is linked to 'b', then 'b' cannot be linked back to 'a', resulting in a strict limit on the presence of non-zero entries.

Imagine a boss assigning tasks to employees; task assignments are one-way. If the boss assigns task A to an employee but that employee doesn’t assign A back, the matrix of relationships (tasks) reflects this one-way tasking.

The next special relation is antisymmetric relation and the requirement here is the following. You want that if both (a, b) and (b, a) are present in your relation, that means if you have a case where an element a is related to b and b is also related to a, then that is possible only if a is equal to b.

In antisymmetric relations, if 'a' relates to 'b' and vice versa, then it can only happen if 'a' equals 'b'. Thus, for distinct elements, both connections cannot exist simultaneously. This condition is pivotal in determining if a relation is antisymmetric.

Consider a hierarchy in a workplace: if a manager (A) interacts with a subordinate (B), the only time they are seen as 'equal' (both connected) is if they share the same role (e.g., two managers). Thus, it shows antisymmetry—relationships cannot be simultaneous at different levels.

So, we have symmetric relation, asymmetric relation and antisymmetric relation. These are the definitions here, and people often wonder that there is some relationship among these three different notions here.

Different types of relations—symmetric, asymmetric, and antisymmetric—are defined by specific characteristics, and understanding these properties helps in the study of relations. It is crucial to grasp how each relation differs and interacts, but fundamentally they fulfill different roles in relational mathematics.

Consider a social network: some users may have mutual connections (symmetric), others may only follow each other one way (asymmetric), and some may only consider similar profiles as friends (antisymmetric). Recognizing these dynamics defines how relationships work in social networks.

Now let us see the last important relation here, which is the transitive relation. And what do we mean by a transitive relation here, so a relation R is called a transitive relation if the following universal quantification is true. We want that if at all a R b and b R c in your relation, then a also should be related to c.

A transitive relation indicates a sequential connection. If 'a' relates to 'b' and 'b' relates to 'c', then 'a' must relate to 'c'. It represents a likelihood that relationships can skip intermediate steps based on other established connections.

Think of a chain of command in a company: if Employee A reports to Manager B, and Manager B reports to Director C, then Employee A can effectively be considered to report indirectly to Director C, demonstrating transitivity.

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