antisümmeetrilisusele

Antisümmeetrilisus, in mathematics, refers to a property of a binary relation where if the relation holds in one direction, it does not hold in the reverse direction, with a special consideration for the case of an element relating to itself. More formally, a relation R on a set A is antisümmeetriline if for any elements a and b in A, whenever a R b and b R a are both true, then it must be that a = b. This means that if two distinct elements are related to each other in both directions, the relation is not antisümmeetriline. However, an element is allowed to be related to itself, i.e., a R a can be true.

A common example of an antisümmeetriline relation is the "less than or equal to" relation (≤) on the set of real numbers. If a ≤ b and b ≤ a are both true, then it logically follows that a must be equal to b. Conversely, the "less than" relation (<) is also antisümmeetriline because if a < b and b < a, this is a contradiction, implying there are no such distinct a and b. The "equal to" relation (=) is also antisümmeetriline, as if a = b and b = a, then a = b is trivially true.

A relation that is both reflexive and antisümmeetriline is called a partial order. Not all relations are antisümmeetriline. For instance, the "is a sibling of" relation is not antisümmeetriline because if person A is a sibling of person B, then person B is also a sibling of person A, and they are not necessarily the same person. Understanding antisümmeetrilisus is fundamental in defining structures like partial orders and is a key concept in set theory and discrete mathematics.