17.4.2. Examples of Antisymmetric Relations

The requirement for an antisymmetric relation is that if both (a, b) and (b, a) are present in the relation, then a must equal b.

An antisymmetric relation is defined with a key condition: if two elements a and b are related such that both (a, b) and (b, a) exist in the relation, then those two elements must be the same, which means a must equal b. This does not restrict the case where (a, b) or (b, a) can exist individually or neither exists—only when both pairs are present must they refer to the same element.

Imagine a situation in a class where students have to partner up for a project. The rule is that a student cannot partner with another student unless they know each other well (known as mutual friends). If student A is known by student B, and student B is known by student A, they can only be partners if they are actually the same student. If they are different, that setup is not allowed.

In terms of matrix properties, in an antisymmetric relation, if we focus on distinct entries ij and ji, only one of those can be 1, which aligns with the antisymmetric definition.

If we visualize an antisymmetric relation using a matrix, we can denote relationships between elements as entries in the matrix. Specifically, if we are checking entries (i, j) and (j, i) for distinct indices i and j, we can have either one of these entries set to 1 or none of them. If both were set to 1, it would contradict the antisymmetric definition, as it would imply that i and j must equal each other, which violates our distinction premise.

Think about the relationships in a social network. If a friend A attributes a claim to friend B and friend B claims the same back to A, this is allowed only if A and B are actually the same person. If they are known differently, you cannot have mutual claims attributed at the same time; there's only one way they can be interconnected.

For instance, the relation R is antisymmetric if it includes the pairs (1, 1) and (2, 2), where both pairs are self-referential and thus satisfy the implication perfectly.

To see antisymmetric relations in practice, consider a set with elements like {1, 2}. If a relation R contains pairs such as (1, 1) or (2, 2), those satisfy antisymmetry as they denote self-relations that are permissible since they refer to the same element. However, if we added distinct pairs like (1, 2) and (2, 1), this would violate the antisymmetry condition as it asserts that both distinct elements are related reciprocally, which cannot happen.

Returning to the classroom analogy, think of it this way: If a student gives credit to themselves (like presenting themselves as a partner on a project), that’s allowed. If student C claims partnership with student D, and D claims the same about C, this would be forbidden unless C and D are indeed the same person. Thus, self-acknowledgment is permitted while reciprocal acknowledgment of distinct identities isn’t under the antisymmetric rule.

Lastly, it is important to note that a relation cannot be both symmetric and antisymmetric unless all elements are self-related (i.e., the relation only includes pairs of the form (a,a)).

Understanding the distinctions between relation types is crucial. A symmetric relation requires that if (a, b) is in the relation, so is (b, a). For a relation to be antisymmetric, having both (a, b) and (b, a) must mean a = b. Thus, for a relation to fit both categories perfectly, it can only include the self-pairings (a, a), leading to an obligatory restriction that means no distinct relationships exist.

Continuing with our social imagery, consider a group of friends. If all friends only acknowledge themselves (like saying 'I am my own best friend'), there’s an indirect mutual acknowledgment, akin to being symmetric with self-pairs. However, if they start claiming friendships that compare distinct individuals, this creates a scenario where the antisymmetric rule kicks in—friendship must remain singular and self-referential to abide by both rules.

--