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Symmetric and antisymmetric relations. By definition, a nonempty relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be related to a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for "is less than or equal to" and "preys on").
Symmetric relations contrast with non-symmetric relations, for which this pair-like behavior is not always observed. An example is the love-relation: if Dave loves Sara then it is possible but not necessary that Sara loves Dave. A special case of non-symmetric relations is asymmetric relations, which only go one way.
An attempt to define the cardinality of a set as the equivalence class of all sets equinumerous to it is problematic in Zermelo–Fraenkel set theory, the standard form of axiomatic set theory, because the equivalence class of any non-empty set would be too large to be a set: it would be a proper class.
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
A relation R is quasitransitive if, and only if, it is the disjoint union of a symmetric relation J and a transitive relation P. [2] J and P are not uniquely determined by a given R; [3] however, the P from the only-if part is minimal. [4] As a consequence, each symmetric relation is quasitransitive, and so is each transitive relation. [5]
A relation R is said to be reflexive if a R a holds for every a in the domain of the relation. Every reflexive relation on a nonempty domain has infinite descending chains, because any constant sequence is a descending chain. For example, in the natural numbers with their usual order ≤, we have 1 ≥ 1 ≥ 1 ≥ ....
Theory Z is a name for various theories of human motivation built on Douglas McGregor's Theory X and Theory Y.Theories X, Y and various versions of Z have been used in human resource management, organizational behavior, organizational communication and organizational development.