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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.
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]
In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. [...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the ...
Similarly, connected relations are sometimes called complete, [7] although this, too, can lead to confusion: The universal relation is also called complete, [8] and "complete" has several other meanings in order theory. Connected relations are also called connex [9] [10] or said to satisfy trichotomy [11] (although the more common definition of ...
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 function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not. A bijection [d] A function that is injective and surjective.