# Asymmetric and antisymmetric relationship

### Antisymmetric relation - Wikipedia

Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric, and/or transitive Which relations in exercise 4 are asymmetric?. A relation R is asymmetric iff it never holds that both R(x, y) and R(y, x). • Every asymmetric Every asymmetric relation is also anti-symmetric. A binary relation is asymmetric if it is antisymmetric and irreflexive over its exact- domain. Instance-Of: Class · Subclass-Of: Antisymmetric-relation, Irreflexive-.

A function is a relation that has exactly one output for every possible input in the domain. The domain does not necessarily have to include all possible objects of a given type. In fact, we sometimes intentionally use a restricted domain in order to satisfy some desirable property.

Properties Of Relations - Anti Symmetric Relation / Sets And Relations - Maths Algebra

The relations discussed above flavors of fruits and fruits of a given flavor are not functions: The main reason for not allowing multiple outputs with the same input is that it lets us apply the same function to different forms of the same thing without changing their equivalence. This is the same as the definition of function, but with the roles of X and Y interchanged; so it means the inverse relation f-1 must also be a function.

In general—regardless of whether or not the original relation was a function—the inverse relation will sometimes be a function, and sometimes not. When f and f-1 are both functions, they are called one-to-one, injective, or invertible functions.

In other words, a surjective function f maps onto every possible output at least once. A function can be neither one-to-one nor onto, both one-to-one and onto in which case it is also called bijective or a one-to-one correspondenceor just one and not the other.

Relations[ edit ] In the above section dealing with functions and their properties, we noted the important property that all functions must have, namely that if a function does map a value from its domain to its co-domain, it must map this value to only one value in the co-domain.

### Asymmetric relation - Oxford Reference

That is the definition of antisymmetric. First off, we need examples of antisymmetric relations. Secondly, pictures most definately do illustrate the concept.

Please tell me why my picture should not be edited back in. Needs a better picture then.

## Oh no, there's been an error

How do I do that? Paul clearly misread the part that says "Must be false if the check mark with the same number z is true for it to be an antisymmetric relation" as saying "Must be false if x is the same number as y for it to be an antisymmetric relation" and you seem to have misunderstood which statement he meant and have edited the other statement such that now it is wrong when originally nothing was wrong. I'll revert the picture. The relation "x is even y is odd" is antisymmetric, but I don't understand how the table helps illustrating it.

It seems that the table tries to illustrate some conditions for an unspecified relation to be antisymmetric but it is very involved and hardly helps understanding. Can it be removed?

## Library Coq.Classes.RelationClasses

Perhaps it would be helpful to instead have an example of something that's not antisymmetric. I guess "the relation x is even, y is odd between a pair x, y of integers is antisymmetric" in the vacuous sensesince there is no pair of integers such that the condition, x is odd, y is even and y is odd, x is even, is true.

If so, maybe this could be made explicit, as that's quite an unintuitive concept in itself.