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A relation can be both symmetric and antisymmetric (in this case, it must be coreflexive), and there are relations which are neither symmetric nor antisymmetric (for example, the "preys on" relation on biological species). Antisymmetry is different from asymmetry: a relation is asymmetric if and only if it is antisymmetric and irreflexive.
For example, the restriction of < from the reals to the integers is still asymmetric, and the converse or dual > of < is also asymmetric. An asymmetric relation need not have the connex property . For example, the strict subset relation ⊊ {\displaystyle \,\subsetneq \,} is asymmetric, and neither of the sets { 1 , 2 } {\displaystyle \{1,2 ...
Antisymmetric for all x, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false). [11] Asymmetric for all x, y ∈ X, if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive. [12]
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").
If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order k {\displaystyle k} may be referred to as a differential k {\displaystyle k} -form , and a completely antisymmetric contravariant tensor field may be referred ...
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric [1]) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition [ 2 ] : p. 38
Antisymmetric wavefunction for a (fermionic) 2-particle state in an infinite square well potential Symmetric wavefunction for a (bosonic) 2-particle state in an infinite square well potential What follows is an example to make the above discussion concrete, using the formalism developed in the article on the mathematical formulation of quantum ...
John- TOP nani-o what- ACC kaimashita bought ka Q John-wa nani-o kaimashita ka John-TOP what-ACC bought Q 'What did John buy' Japanese has an overt "question particle" (ka), which appears at the end of the sentence in questions. It is generally assumed that languages such as English have a "covert" (i.e. phonologically null) equivalent of this particle in the 'C' position of the clause — the ...