nonantisymmetric

A nonantisymmetric relation is a type of binary relation on a set that does not satisfy the property of antisymmetry. Antisymmetry, in its strict definition, means that if a relation holds from an element x to an element y, and also from y to x, then x must be equal to y. A relation is nonantisymmetric if there exist distinct elements x and y in the set such that the relation holds from x to y and also from y to x.

For example, consider the relation "is a sibling of" on the set of people. If person A is a sibling of person B, then person B is also a sibling of person A. Since A and B can be distinct individuals, this relation is nonantisymmetric. Another example is the "is indebted to" relation. If person X owes money to person Y, it is possible for person Y to also owe money to person X, where X and Y are different people.

In contrast, a symmetric relation is one where if x is related to y, then y is also related to x. Antisymmetry is a stronger condition than symmetry. A relation can be symmetric without being antisymmetric (like "is a sibling of"). It can also be antisymmetric without being symmetric (like "is less than" on numbers, where if x < y, then y is not < x). A nonantisymmetric relation simply fails to meet the specific requirement of antisymmetry, meaning it's not guaranteed that if a relation goes both ways between two distinct elements, it implies those elements are the same.