Asymmetrical relation

A two-digit relation on a set is called asymmetric if there is no pair for which the converse also applies. ${\ displaystyle R}$ ${\ displaystyle (x, y)}$ ${\ displaystyle xRy}$ ${\ displaystyle yRx}$

Asymmetry is one of the prerequisites for an (irreflexive) strict order .

definition

Is a quantity and a two-digit relation to , it means asymmetric when ${\ displaystyle M}$ ${\ displaystyle R \ subseteq M \ times M}$ ${\ displaystyle M}$ ${\ displaystyle R}$

{\ displaystyle \ forall x, y \ in M: xRy \ Rightarrow \ neg (yRx)}

applies.

Not symmetrical relation

If there is a relation that is not symmetric , then there is at least one pair for which the inverse relation does not hold; so applies ${\ displaystyle R}$ ${\ displaystyle R ^ {- 1}}$

{\ displaystyle \ exists x, y \ in M: xRy \ land \ neg (yRx)}

.

A non-empty asymmetric relation is never symmetric. An asymmetrical relation is also always irreflexive . The concept of antisymmetry , which also allows reflexivity, must be distinguished from asymmetry . An asymmetrical relation is therefore a special case of an antisymmetrical relation.

There are relations that are neither symmetrical nor antisymmetrical, and certainly not asymmetrical. An example is provided by the definition on the natural numbers. ${\ displaystyle xRy: \ Leftrightarrow x> 2}$

Examples

Are asymmetrical

the relation “is (real) smaller than” on the real numbers , which is also a strict total order . The same applies to the relation “is (real) greater than”. ${\ displaystyle <}$ ${\ displaystyle> \}$

the relation “is a real subset of” and also the relation “is a real superset of” as relationships between sets. In a system of sets or of subsets of a given set, they are also a strict partial order . ${\ displaystyle \ subset}$ ${\ displaystyle \ supset}$

properties

Every asymmetrical relation is a non-symmetrical relation and also an antisymmetrical relation .

The intersection of an asymmetrical relation and its converse relation is always empty, they are disjoint: ${\ displaystyle R}$ ${\ displaystyle R ^ {- 1}}$
${\ displaystyle R \ cap R ^ {- 1} = \ emptyset}$

Every subset of an asymmetric relation is again asymmetric.

Remarks

↑ see also: Ingmar Lehmann , Wolfgang Schulz: Sets - Relations - Functions. A clear introduction. 3rd, revised and expanded edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0162-3 , p. 64 f.

[1] see also: Ingmar Lehmann , Wolfgang Schulz: Sets - Relations - Functions. A clear introduction. 3rd, revised and expanded edition. Teubner, Wiesbaden 2007, ISBN 978-3-8351-0162-3 , p. 64 f.