17.4. Antisymmetric Relations

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. Contra-positively, if a is not equal to b, then you can have either (a, b) present in the relation or (b, a) present in the relation or none of them being present in the relation.

An antisymmetric relation has a special property: if one element is related to another (for example, a is related to b), and vice versa (b is related to a), then the only way this can happen is if the two elements are actually the same (a = b). In simpler words, for two different elements, you cannot have both a related to b and b related to a at the same time. Thus, if a and b are distinct, at least one of these relationships must not hold.

Imagine you are defining relationships in a small group of friends. If Alice says she is friends with Bob, and Bob says he’s friends with Alice, that’s fine, but if you say Alice and Bob are not the same person (they are distinct), they cannot both say they are friends with each other because that would imply they are the same person. So, in our group, friendships don't work that way—they must respect individuality.

In terms of matrix properties, if you focus on the ith row and jth column where i and j are distinct, then only one of those entries can be 1. Of course, both of them can be 0, that is also fine, because that means that neither a R b nor b R a. The condition demands that if at all a and b are present in the relation, then that is possible only when a and b are the same. For distinct elements, you cannot have both (a, b) and (b, a) present in your relation.

In a matrix that represents an antisymmetric relation, if we look at two different elements (let's say a and b), we cannot have both relationships (a, b) and (b, a) marked as true (denoted by 1) in the matrix. This means if we see a 1 in the cell for (a, b), the cell for (b, a) must contain a 0, complying with the antisymmetry rule. If both are 0, it simply means there’s no direct relation.

Think about a directed highway system. If there’s a one-way street from point A to point B, there cannot be a one-way road going back from B to A using the same road; it upholds the idea of clarity in direction. Two points cannot point to each other unless they are the same point—there simply cannot be mutual connections without intersection.

Here are some examples: relation R is an antisymmetric relation because you have (a, b) present here namely (1, 1) and you also have (b, a), present here namely (1, 1), but the implication should be that 1 equals 1 which is true. R is not an example of antisymmetric relation because you have a case here namely you have distinct (a, b) such that both (a, b) as well as (b, a) are present in your relation.

Let’s look at some specific relations: If we have (1, 1) present in a relation, that satisfies antisymmetry because both elements are the same. However, if we have (2, 1) and (1, 2), it violates the antisymmetric property because we have two different values that are informing a mutual relationship, which contradicts the rule of antisymmetry.

Imagine a situation where you're keeping track of who owes money to whom. If Alice owes Bob $10, and Bob also owes Alice$10, then they can't be two distinct parties in this scenario of debt—either they need to just settle or both must be considered as one unit in this financial transaction. The concept thus prevents ambiguity in such transactions.

Relation R is also an example of universal quantification because you have (a, b) present here, but the (b, a) is not present in the relation R, that means the premise of this implication is vacuously true for R and that is why this R is not violating this universal quantification.

In cases where a relation consists of no pairs or only a few that do not adhere to the antisymmetric requirements, we can say that it vacuously satisfies the antisymmetric condition. When we don't have contrasting pairs, the rule doesn't find a basis to apply and hence holds true by default.

Consider a classroom setting where no students have a comparative relationship (like the absence of conflicts or favoritism). When there are no instances of competition or rivalry, it simply follows that there can’t be an overlap or contradiction. Hence, in such empty atmospheres, underlying rules are adhered to rather seamlessly.

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